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Your data matches 579 different statistics following compositions of up to 3 maps.
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Matching statistic: St000455
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000455: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000455: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,5,6,4,7] => ([(2,3),(4,6),(5,6)],7)
=> 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000089
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00294: Standard tableaux āpeak compositionā¶ Integer compositions
St000089: Integer compositions ā¶ ā¤Result quality: 53% āvalues known / values provided: 53%ādistinct values known / distinct values provided: 100%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00294: Standard tableaux āpeak compositionā¶ Integer compositions
St000089: Integer compositions ā¶ ā¤Result quality: 53% āvalues known / values provided: 53%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [2] => 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [5] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [4] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [3,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [3,2,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [3,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [5,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [4,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [7] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [6] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [2,3,3] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [4,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [6] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [5] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [4] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [3] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [3,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [3,3,4,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> [3,3,3,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> [3,2,4,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [3,2,3,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [3,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> [3,2,2,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [3,2,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> [5,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> [5,4,4] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> [5,3,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> [4,4,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> [4,3,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [4,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [4,2,5] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> [4,2,4] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [7,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [6,5] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [6,4] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [6,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> [5,5] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [5,4] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [5,3] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [9] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [8] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [7] => 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> [2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> [2,3,4,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> [2,3,3,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [2,2,4,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> [2,2,3,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [4,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [3,3,5] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [3,3,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [3,2,5] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [3,2,4] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [3,2,3] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> [5,5] => ? = 0
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [5,4] => ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [4,5] => ? = 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [4,4] => ? = 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [4,3] => 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [8] => ? = 0
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [7] => 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [6] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [5] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 0
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [3,4] => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [3,3] => 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [7] => 0
Description
The absolute variation of a composition.
Matching statistic: St000022
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000022: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000022: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [10,1,4,6,2,7,9,3,5,8] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,5,1,6,8,2,4,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [4,1,2,10,6,7,9,3,5,8] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,1,10,6,2,3,9,4,5,8] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,7,6,3,9,4,5,8] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [9,1,6,5,2,8,3,4,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,3,6,7,9,4,5,8] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,1,4,5,2,10,9,3,7,8] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,3,4,1,9,8,2,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,10,9,3,7,8] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,6,5,2,3,9,4,7,8] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,2,5,6,3,9,4,7,8] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [9,1,4,5,2,8,3,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,10,9,4,7,8] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,9,8,3,6,7] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,9,5,2,3,4,6,7,8] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,2,5,9,3,4,6,7,8] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,1,2,3,9,10,4,6,7,8] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,1,2,8,9,3,5,6,7] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [9,1,4,6,2,7,8,3,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [8,3,5,1,6,7,2,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,1,2,9,6,7,8,3,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [6,8,5,1,2,7,3,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [9,1,6,5,2,3,8,4,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [9,1,2,5,6,3,8,4,7] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,1,2,3,6,9,8,4,7] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000317
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000317: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000317: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10,15] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6,13,14] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> [10,6,11,4,7,2,5,12,13,1,3,8,9,14] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8,12,13] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8,12] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11,12] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> [10,6,3,11,12,2,7,8,13,1,4,5,9,14] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> [10,6,3,11,12,2,7,8,13,1,4,5,9] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5,12,13] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> [9,5,3,10,2,6,11,12,1,4,7,8,13] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11,12] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [8,4,3,2,9,10,11,1,5,6,7,12] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9,10] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> [10,11,6,7,12,3,4,8,13,1,2,5,9,14] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> [10,11,6,7,12,3,4,8,13,1,2,5,9] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5,12,13] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7,12] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9,10] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9,10] => ? = 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St000355
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000355: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000355: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [11,10,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [10,9,7,8,5,6,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [11,10,9,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [10,9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [10,9,8,6,7,1,2,3,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [10,9,6,7,8,1,2,3,4,5] => ? = 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000405
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000405: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000405: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [10,1,4,6,2,7,9,3,5,8] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [9,3,5,1,6,8,2,4,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [4,1,2,10,6,7,9,3,5,8] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,1,10,6,2,3,9,4,5,8] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,7,6,3,9,4,5,8] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [9,1,6,5,2,8,3,4,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [10,1,2,3,6,7,9,4,5,8] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [8,1,4,5,7,2,3,6] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,1,4,5,2,10,9,3,7,8] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,3,4,1,9,8,2,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,10,9,3,7,8] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,6,5,2,3,9,4,7,8] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [10,1,2,5,6,3,9,4,7,8] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [9,1,4,5,2,8,3,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,10,9,4,7,8] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,9,8,3,6,7] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,9,5,2,3,4,6,7,8] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,1,2,5,9,3,4,6,7,8] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,1,2,3,9,10,4,6,7,8] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,1,2,8,9,3,5,6,7] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [9,1,4,6,2,7,8,3,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [8,3,5,1,6,7,2,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,1,2,9,6,7,8,3,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [6,8,5,1,2,7,3,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [9,1,6,5,2,3,8,4,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [9,1,2,5,6,3,8,4,7] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [8,1,4,5,2,7,3,6] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,1,2,3,6,9,8,4,7] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000406
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000406: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000406: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [11,10,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [10,9,7,8,5,6,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [11,10,9,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [10,9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [10,9,8,6,7,1,2,3,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [10,9,6,7,8,1,2,3,4,5] => ? = 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
Description
The number of occurrences of the pattern 3241 in a permutation.
Matching statistic: St000407
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000407: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000407: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [11,10,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [10,9,7,8,5,6,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [11,10,9,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [10,9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [10,9,8,6,7,1,2,3,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [10,9,6,7,8,1,2,3,4,5] => ? = 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
Description
The number of occurrences of the pattern 2143 in a permutation.
A permutation $\pi$ avoids this pattern if and only if it is ''vexillary'' as introduced in [1].
Matching statistic: St000425
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000425: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000425: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [11,10,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [10,9,7,8,5,6,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [11,10,9,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [10,9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [10,9,8,6,7,1,2,3,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [10,9,6,7,8,1,2,3,4,5] => ? = 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
Matching statistic: St000663
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000663: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000663: Permutations ā¶ ā¤Result quality: 47% āvalues known / values provided: 47%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [11,10,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [10,9,7,8,5,6,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [11,10,9,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [10,9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [10,9,8,6,7,1,2,3,4,5] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [10,9,6,7,8,1,2,3,4,5] => ? = 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
Description
The number of right floats of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
The following 569 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000908The length of the shortest maximal antichain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001735The number of permutations with the same set of runs. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000750The number of occurrences of the pattern 4213 in a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000056The decomposition (or block) number of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001344The neighbouring number of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000570The Edelman-Greene number of a permutation. St000914The sum of the values of the Mƶbius function of a poset. St001162The minimum jump of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001625The Mƶbius invariant of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000741The Colin de VerdiĆØre graph invariant. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001890The maximum magnitude of the Mƶbius function of a poset. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000546The number of global descents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000007The number of saliances of the permutation. St000842The breadth of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000929The constant term of the character polynomial of an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001593This is the number of standard Young tableaux of the given shifted shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001845The number of join irreducibles minus the rank of a lattice. St000781The number of proper colouring schemes of a Ferrers diagram. St001568The smallest positive integer that does not appear twice in the partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000264The girth of a graph, which is not a tree. St000068The number of minimal elements in a poset. St000352The Elizalde-Pak rank of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001434The number of negative sum pairs of a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St001429The number of negative entries in a signed permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001555The order of a signed permutation. St000214The number of adjacencies of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000408The number of occurrences of the pattern 4231 in a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000787The number of flips required to make a perfect matching noncrossing. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000630The length of the shortest palindromic decomposition of a binary word. St000862The number of parts of the shifted shape of a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000983The length of the longest alternating subword. St001330The hat guessing number of a graph. St000210Minimum over maximum difference of elements in cycles. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000054The first entry of the permutation. St000260The radius of a connected graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000627The exponent of a binary word. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000031The number of cycles in the cycle decomposition of a permutation. St000383The last part of an integer composition. St001260The permanent of an alternating sign matrix. St000381The largest part of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001947The number of ties in a parking function. St000065The number of entries equal to -1 in an alternating sign matrix. St000069The number of maximal elements of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000237The number of small exceedances. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000366The number of double descents of a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000629The defect of a binary word. St001394The genus of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000374The number of exclusive right-to-left minima of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000717The number of ordinal summands of a poset. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001618The cardinality of the Frattini sublattice of a lattice. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001513The number of nested exceedences of a permutation. St001536The number of cyclic misalignments of a permutation. St001728The number of invisible descents of a permutation. St001665The number of pure excedances of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000233The number of nestings of a set partition. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000496The rcs statistic of a set partition. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001115The number of even descents of a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000883The number of longest increasing subsequences of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000701The protection number of a binary tree. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001616The number of neutral elements in a lattice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000382The first part of an integer composition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000894The trace of an alternating sign matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001524The degree of symmetry of a binary word. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000295The length of the border of a binary word. St000657The smallest part of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001267The length of the Lyndon factorization of the binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001569The maximal modular displacement of a permutation. St001884The number of borders of a binary word. St001885The number of binary words with the same proper border set. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000905The number of different multiplicities of parts of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000735The last entry on the main diagonal of a standard tableau. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000002The number of occurrences of the pattern 123 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000095The number of triangles of a graph. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000313The number of degree 2 vertices of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000338The number of pixed points of a permutation. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000389The number of runs of ones of odd length in a binary word. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000628The balance of a binary word. St000649The number of 3-excedences of a permutation. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000693The modular (standard) major index of a standard tableau. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000921The number of internal inversions of a binary word. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001430The number of positive entries in a signed permutation. St001433The flag major index of a signed permutation. St001485The modular major index of a binary word. St001520The number of strict 3-descents. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001721The degree of a binary word. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001819The flag Denert index of a signed permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001871The number of triconnected components of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001948The number of augmented double ascents of a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000096The number of spanning trees of a graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000390The number of runs of ones in a binary word. St000450The number of edges minus the number of vertices plus 2 of a graph. St000545The number of parabolic double cosets with minimal element being the given permutation. St000654The first descent of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000805The number of peaks of the associated bargraph. St000843The decomposition number of a perfect matching. St000847The number of standard Young tableaux whose descent set is the binary word. St000864The number of circled entries of the shifted recording tableau of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000924The number of topologically connected components of a perfect matching. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000958The number of Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001081The number of minimal length factorizations of a permutation into star transpositions. St001096The size of the overlap set of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001470The cyclic holeyness of a permutation. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001518The number of graphs with the same ordinary spectrum as the given graph. St001589The nesting number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001684The reduced word complexity of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001765The number of connected components of the friends and strangers graph. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001828The Euler characteristic of a graph. St001838The number of nonempty primitive factors of a binary word. St001851The number of Hecke atoms of a signed permutation. St001853The size of the two-sided Kazhdan-Lusztig cell, St001854The size of the left Kazhdan-Lusztig cell, St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000758The length of the longest staircase fitting into an integer composition. St000822The Hadwiger number of the graph. St000903The number of different parts of an integer composition. St000942The number of critical left to right maxima of the parking functions. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001415The length of the longest palindromic prefix of a binary word. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001893The flag descent of a signed permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000402Half the size of the symmetry class of a permutation. St000498The lcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000011The number of touch points (or returns) of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000729The minimal arc length of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000920The logarithmic height of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000635The number of strictly order preserving maps of a poset into itself. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001910The height of the middle non-run of a Dyck path. St000782The indicator function of whether a given perfect matching is an L & P matching. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000647The number of big descents of a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order.
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