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Your data matches 82 different statistics following compositions of up to 3 maps.
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Matching statistic: St000390
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Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00114: Permutations āconnectivity setā¶ Binary words
St000390: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00114: Permutations āconnectivity setā¶ Binary words
St000390: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 0 => 0
[1,1,0,0]
=> [1,2] => 1 => 1
[1,0,1,0,1,0]
=> [3,2,1] => 00 => 0
[1,0,1,1,0,0]
=> [2,3,1] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => 00 => 0
[1,1,0,1,0,0]
=> [2,1,3] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 11 => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 000 => 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 001 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 011 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0000 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 0000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 0001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0000 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 0001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 0001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 0001 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0000 => 0
Description
The number of runs of ones in a binary word.
Matching statistic: St000929
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(load all 4 compositions to match this statistic)
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
Mp00040: Integer compositions āto partitionā¶ Integer partitions
Mp00322: Integer partitions āLoehr-Warringtonā¶ Integer partitions
St000929: Integer partitions ā¶ ā¤Result quality: 99% āvalues known / values provided: 99%ādistinct values known / distinct values provided: 100%
Mp00040: Integer compositions āto partitionā¶ Integer partitions
Mp00322: Integer partitions āLoehr-Warringtonā¶ Integer partitions
St000929: Integer partitions ā¶ ā¤Result quality: 99% āvalues known / values provided: 99%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1]
=> [2]
=> 0
[1,1,0,0]
=> [2] => [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [2,1]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [3]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [3]
=> 0
[1,1,0,1,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [3,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [2,2]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [2,2]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [4]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [3,2]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,2,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,2,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [5]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [5]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [4,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [5]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [4,1]
=> 0
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? => ?
=> ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ?
=> ?
=> ? = 0
[1,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? => ?
=> ?
=> ? = 1
[1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? => ?
=> ?
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ?
=> ?
=> ? = 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001353
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St001353: Graphs ā¶ ā¤Result quality: 98% āvalues known / values provided: 98%ādistinct values known / distinct values provided: 100%
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St001353: Graphs ā¶ ā¤Result quality: 98% āvalues known / values provided: 98%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => ([],2)
=> 0
[1,1,0,0]
=> [2] => [1] => ([],1)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,0]
=> [3] => [1] => ([],1)
=> 1
[1,1,1,0,0,0]
=> [3] => [1] => ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => ([],2)
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => ([],8)
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1] => [9] => ([],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ([],10)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ([],11)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => [1,7] => ([(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1,1] => [5,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1,1] => [4,1,4] => ([(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => [1,8] => ([(7,8)],9)
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,1,1,2] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => ([(6,7)],8)
=> ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => ([(6,7)],8)
=> ? = 0
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [1,9] => ([(8,9)],10)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,3,1,1] => [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,3,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? => ? => ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? => ? => ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ? => ?
=> ? = 0
Description
The number of prime nodes in the modular decomposition of a graph.
Matching statistic: St001356
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St001356: Graphs ā¶ ā¤Result quality: 98% āvalues known / values provided: 98%ādistinct values known / distinct values provided: 100%
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St001356: Graphs ā¶ ā¤Result quality: 98% āvalues known / values provided: 98%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => ([],2)
=> 0
[1,1,0,0]
=> [2] => [1] => ([],1)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,0]
=> [3] => [1] => ([],1)
=> 1
[1,1,1,0,0,0]
=> [3] => [1] => ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => ([],2)
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => ([],1)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => ([],8)
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1] => [9] => ([],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ([],10)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ([],11)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => [1,7] => ([(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1,1] => [5,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1,1] => [4,1,4] => ([(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => [1,8] => ([(7,8)],9)
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,1,1,2] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => ([(6,7)],8)
=> ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => ([(6,7)],8)
=> ? = 0
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [1,9] => ([(8,9)],10)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,3,1,1] => [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,3,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? => ? => ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? => ? => ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? => ? => ?
=> ? = 1
[1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? => ? => ?
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? => ? => ?
=> ? = 0
Description
The number of vertices in prime modules of a graph.
Matching statistic: St000326
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
Mp00131: Permutations ādescent bottomsā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 86% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 100%
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
Mp00131: Permutations ādescent bottomsā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 86% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 2 = 1 + 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 10 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 10 => 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 01 => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => 01 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => 100 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 101 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,4,1,3] => 100 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,4,3,1] => 101 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,1,4,2] => 110 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 100 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => 110 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => 011 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 011 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => 110 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 011 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => 011 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 011 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 1011 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 1010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 1010 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 1001 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 1010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 1011 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 1010 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 1001 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 1011 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 1010 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 1001 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 1011 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,1,5,4,2] => 1101 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,5,1,4,2] => 1100 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 1101 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,4,2] => 1100 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,5,4,1,2] => 1001 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 1101 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => 1100 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 1101 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 0111 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => 0111 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => 1101 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 0111 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 0111 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,1,5,3,2] => 1110 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6,8] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6,8] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6,8] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6,8] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6,8] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,8,6,7] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,7,8,6] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,1,4,7,8,6] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,3,5,7,4,8,6] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,4,6,8,7] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,6,8,7] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,1,5,6,4,8,7] => ? => ? => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,5,1,6,7,4,8] => ? => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [2,1,6,3,4,7,5,8] => ? => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5,8] => ? => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,1,7,4,5,8] => ? => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ? => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [2,3,6,1,7,4,8,5] => ? => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,8,5,7] => ? => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,3,7,4,5,8,6] => ? => ? => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,4,6,1,2,8,5,7] => ? => ? => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5,7,8] => ? => ? => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5,7,8] => ? => ? => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,7,2,5,8] => ? => ? => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,4,1,7,2,8,5,6] => ? => ? => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,7,1,2,8,5,6] => ? => ? => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,7,8,2,5,6] => ? => ? => ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,7,2,8,6] => ? => ? => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,1,4,2,5,6,8,7] => ? => ? => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [3,5,1,6,2,8,4,7] => ? => ? => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4,7,8] => ? => ? => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [3,5,7,1,8,2,4,6] => ? => ? => ? = 0 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,5,4,7,8,6] => ? => ? => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [3,5,8,1,2,4,6,7] => ? => ? => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,8,6,7] => ? => ? => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,8,6,7] => ? => ? => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ? => ? => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,3,5,2,4,6,8,7] => ? => ? => ? = 1 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,5,6,7,8,4] => ? => ? => ? = 1 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
Mp00114: Permutations āconnectivity setā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 84% āvalues known / values provided: 84%ādistinct values known / distinct values provided: 100%
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
Mp00114: Permutations āconnectivity setā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 84% āvalues known / values provided: 84%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => 0 => 0
[1,1,0,0]
=> [1,2] => [1,2] => 1 => 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 01 => 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 00 => 0
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 10 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 010 => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 010 => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,4,1,3] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,4,3,1] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,1,4,2] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => 000 => 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 100 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => 000 => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => 100 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 0100 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 0100 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 0100 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 0100 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 0000 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 0100 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 0000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 0000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,1,5,4,2] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,5,1,4,2] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,4,2] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,5,4,1,2] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => 1000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => 0000 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 1000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 1000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 1000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,1,5,3,2] => 0000 => 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6,8] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6,8] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6,8] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6,8] => ? => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6,8] => ? => ? => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,8,6,7] => ? => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7,8] => ? => ? => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ? => ? => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7,8] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6,8] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6,8] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6,8] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,7,8,6] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,1,4,7,8,6] => ? => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,3,5,7,4,8,6] => ? => ? => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,4,6,8,7] => ? => ? => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,6,8,7] => ? => ? => ? = 0
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,1,5,6,4,8,7] => ? => ? => ? = 0
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,5,1,6,7,4,8] => ? => ? => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [2,1,6,3,4,7,5,8] => ? => ? => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5,8] => ? => ? => ? = 0
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,1,7,4,5,8] => ? => ? => ? = 0
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ? => ? => ? = 0
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [2,3,6,1,7,4,8,5] => ? => ? => ? = 0
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,8,5,7] => ? => ? => ? = 0
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,3,7,4,5,8,6] => ? => ? => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,4,6,1,2,8,5,7] => ? => ? => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5,7,8] => ? => ? => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5,7,8] => ? => ? => ? = 0
[1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,7,2,5,8] => ? => ? => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,4,1,7,2,8,5,6] => ? => ? => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,7,1,2,8,5,6] => ? => ? => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,7,8,2,5,6] => ? => ? => ? = 0
[1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,7,2,8,6] => ? => ? => ? = 0
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,1,4,2,5,6,8,7] => ? => ? => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [3,5,1,6,2,8,4,7] => ? => ? => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4,7,8] => ? => ? => ? = 0
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [3,5,7,1,8,2,4,6] => ? => ? => ? = 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,5,4,7,8,6] => ? => ? => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [3,5,8,1,2,4,6,7] => ? => ? => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,8,6,7] => ? => ? => ? = 0
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,8,6,7] => ? => ? => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ? => ? => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,3,5,2,4,6,8,7] => ? => ? => ? = 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,5,6,7,8,4] => ? => ? => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000745
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00101: Dyck paths ādecomposition reverseā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 80% āvalues known / values provided: 80%ādistinct values known / distinct values provided: 100%
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 80% āvalues known / values provided: 80%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [[1,2]]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [[1,2],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [[1,3],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [5,6,1,7,2,8,3,4] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> [5,6,1,2,7,3,4,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> [4,6,1,2,7,3,5,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,1,2,6,3,5,7,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,1,5,7,2,3,6,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,1,6,2,4,5,7,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,7,1,8,2,4,6] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [3,1,6,7,8,2,4,5] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,5,1,6,2,8,3,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,5,1,6,7,2,3,8] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [3,6,1,2,7,4,5,8] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,1,5,2,4,6,7,8] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6,8] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,6,1,2,5,7,8] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,4,1,7,8,2,5,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,6,7,8,3,5] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [3,1,6,2,7,8,4,5] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [3,1,2,6,4,8,5,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,4,7,1,2,8,5,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,4,5,7,2,8,3,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,4,1,7,2,8,5,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,8,5,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,1,5,6,7,2,3,8] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [3,4,5,1,8,2,6,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,5,7,1,8,2,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [3,4,5,1,7,2,6,8] => ?
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,6,1,7,2,3,8,5] => ?
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,5,1,2,3,6,8,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,6,1,2,7,3,8,5] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [4,5,1,2,7,3,8,6] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> [4,1,5,2,3,6,8,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [4,1,5,7,2,3,8,6] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [1,5,2,6,7,3,8,4] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [3,4,5,1,7,2,8,6] => ?
=> ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0]
=> [4,6,1,2,3,7,5,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,1,5,6,2,7,4,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [3,4,5,1,2,7,6,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,5,1,2,6,3,7,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [3,1,2,4,6,5,7,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,5,1,2,6,4,7,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,4,7,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,5,4,6,7,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,6,7,1,3,4,5,8] => ?
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6,8] => ?
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,1,6,3,4,5,7,8] => ?
=> ? = 1 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000352
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
St000352: Permutations ā¶ ā¤Result quality: 73% āvalues known / values provided: 73%ādistinct values known / distinct values provided: 100%
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
St000352: Permutations ā¶ ā¤Result quality: 73% āvalues known / values provided: 73%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,2,3,7,4,6] => ? = 0
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,3,6,4,7] => ? = 0
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4,7] => ? = 0
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,2,6,3,7,4] => ? = 0
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,2,6,3,7,4] => ? = 0
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6] => ? = 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,1,2,6,3,5,7] => ? = 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6] => ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [5,1,2,6,3,7,4] => ? = 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,2,6,3,7,5] => ? = 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [5,1,2,6,3,7,4] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [4,1,2,5,3,7,6] => ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,7,6] => ? = 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,1,2,7] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,1,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => ? = 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,1,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => ? = 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,7,5] => ? = 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,5,2,3,7,6] => ? = 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,7,6] => ? = 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,5,3,1,2,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => ? = 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [5,3,4,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,2,1,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => ? = 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [5,1,6,2,7,3,4] => ? = 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,6,2,7,3,5] => ? = 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,2,3,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [5,1,6,2,7,3,4] => ? = 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,4,1,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,5,2,7,3,6] => ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => ? = 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [5,1,6,2,7,3,4] => ? = 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,6,2,7,3,5] => ? = 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? = 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [5,1,6,2,7,3,4] => ? = 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3,7] => ? = 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000007
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00159: Permutations āDemazure product with inverseā¶ Permutations
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
St000007: Permutations ā¶ ā¤Result quality: 70% āvalues known / values provided: 70%ādistinct values known / distinct values provided: 100%
Mp00159: Permutations āDemazure product with inverseā¶ Permutations
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
St000007: Permutations ā¶ ā¤Result quality: 70% āvalues known / values provided: 70%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [2,3,1,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,1,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [2,3,1,4] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => [3,2,4,1,5] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => [3,2,4,1,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,2,4,1] => [3,2,4,1,5] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => [2,3,4,1,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => [2,3,1,5,4] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,1,5,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,3,2,1] => [4,3,2,1,5] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [7,5,3,4,2,6,1] => [5,3,4,2,6,1,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [7,5,3,4,2,6,1] => [5,3,4,2,6,1,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [7,5,3,4,2,6,1] => [5,3,4,2,6,1,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [7,5,3,4,2,6,1] => [5,3,4,2,6,1,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,5,3,4,2,6,1] => [5,3,4,2,6,1,7] => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,7,1] => [7,5,3,4,2,6,1] => [5,3,4,2,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [7,3,2,4,5,6,1] => [3,2,4,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [7,3,2,4,5,6,1] => [3,2,4,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [7,5,3,4,2,6,1] => [5,3,4,2,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [7,4,3,2,5,6,1] => [4,3,2,5,6,1,7] => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [7,3,2,4,5,6,1] => [3,2,4,5,6,1,7] => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,1,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [6,3,2,4,5,1,7] => [3,2,4,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [6,3,2,4,5,1,7] => [3,2,4,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,1,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [6,3,2,4,5,1,7] => [3,2,4,5,1,7,6] => ? = 1 + 1
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,1,2,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,1,2,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,1,2,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,1,2,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,1,2,7] => [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => [4,3,2,1,6,7,5] => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,1,6,7] => [5,4,3,2,1,6,7] => [4,3,2,1,6,7,5] => ? = 1 + 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,1,6,7] => [5,4,3,2,1,6,7] => [4,3,2,1,6,7,5] => ? = 1 + 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,1,6,7] => [5,4,3,2,1,6,7] => [4,3,2,1,6,7,5] => ? = 1 + 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => [5,4,3,2,1,6,7] => [4,3,2,1,6,7,5] => ? = 1 + 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,1,6,7] => [5,4,3,2,1,6,7] => [4,3,2,1,6,7,5] => ? = 1 + 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,1,6,7] => [5,4,3,2,1,6,7] => [4,3,2,1,6,7,5] => ? = 1 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000980
Mp00100: Dyck paths ātouch compositionā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St000980: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 66%ādistinct values known / distinct values provided: 50%
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St000980: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 66%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [2] => [1] => [1,0]
=> ? = 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? = 1
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$.
We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.
The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $eA$. St000481The number of upper covers of a partition in dominance order. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000315The number of isolated vertices of a graph. St000948The chromatic discriminant of a graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001271The competition number of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000546The number of global descents of a permutation. St000234The number of global ascents of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. 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. St000678The number of up steps after the last double rise of a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000456The monochromatic index of a connected graph. 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. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000096The number of spanning trees of a graph. St000237The number of small exceedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001367The smallest number which does not occur as degree of a vertex in a graph. St001691The number of kings in a graph. St000056The decomposition (or block) number of a permutation. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St000991The number of right-to-left minima of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000552The number of cut vertices of a graph. St000917The open packing number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001545The second Elser number of a connected graph. St000259The diameter of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000221The number of strong fixed points of a permutation. St000461The rix statistic of a permutation. St000873The aix statistic of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001728The number of invisible descents of a permutation. St001948The number of augmented double ascents of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$.
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