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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000093
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => ([],1)
=> 1
[[1,2]]
=> 0 => [2] => ([],2)
=> 2
[[1],[2]]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> 00 => [3] => ([],3)
=> 3
[[1,3],[2]]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[[1,2],[3]]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[3]]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> 000 => [4] => ([],4)
=> 4
[[1,3,4],[2]]
=> 100 => [1,3] => ([(2,3)],4)
=> 3
[[1,2,4],[3]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[[1,2,3],[4]]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,3],[2,4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[[1,4],[2],[3]]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> 0000 => [5] => ([],5)
=> 5
[[1,3,4,5],[2]]
=> 1000 => [1,4] => ([(3,4)],5)
=> 4
[[1,2,4,5],[3]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 4
[[1,2,3,5],[4]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[[1,2,3,4],[5]]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[3,4]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 4
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,3],[4,5]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => ([],6)
=> 6
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => ([(4,5)],6)
=> 5
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 5
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 5
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> 0 = 1 - 1
[[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
[[1],[2]]
=> [[1,2]]
=> 0 = 1 - 1
[[1,2,3]]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[[1,3],[2]]
=> [[1,2],[3]]
=> 1 = 2 - 1
[[1,2],[3]]
=> [[1,3],[2]]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [[1,2,3]]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3 = 4 - 1
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 3 = 4 - 1
[[1,4],[2,5],[3,6],[7],[8],[9]]
=> ?
=> ? = 2 - 1
[[1,4],[2,5],[3],[6],[7],[8],[9]]
=> ?
=> ? = 2 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000786
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => ([],1)
=> 1
[[1,2]]
=> 0 => [2] => ([],2)
=> 2
[[1],[2]]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> 00 => [3] => ([],3)
=> 3
[[1,3],[2]]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[[1,2],[3]]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[3]]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> 000 => [4] => ([],4)
=> 4
[[1,3,4],[2]]
=> 100 => [1,3] => ([(2,3)],4)
=> 3
[[1,2,4],[3]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[[1,2,3],[4]]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,3],[2,4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[[1,4],[2],[3]]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> 0000 => [5] => ([],5)
=> 5
[[1,3,4,5],[2]]
=> 1000 => [1,4] => ([(3,4)],5)
=> 4
[[1,2,4,5],[3]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 4
[[1,2,3,5],[4]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[[1,2,3,4],[5]]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[3,4]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 4
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,3],[4,5]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => ([],6)
=> 6
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => ([(4,5)],6)
=> 5
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 5
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 5
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 011111111 => [2,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 111111111 => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St000288
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00278: Binary words —rowmotion⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => => => ? = 1 - 1
[[1,2]]
=> 0 => 0 => 1 => 1 = 2 - 1
[[1],[2]]
=> 1 => 1 => 0 => 0 = 1 - 1
[[1,2,3]]
=> 00 => 00 => 11 => 2 = 3 - 1
[[1,3],[2]]
=> 10 => 01 => 10 => 1 = 2 - 1
[[1,2],[3]]
=> 01 => 10 => 01 => 1 = 2 - 1
[[1],[2],[3]]
=> 11 => 11 => 00 => 0 = 1 - 1
[[1,2,3,4]]
=> 000 => 000 => 111 => 3 = 4 - 1
[[1,3,4],[2]]
=> 100 => 001 => 110 => 2 = 3 - 1
[[1,2,4],[3]]
=> 010 => 100 => 011 => 2 = 3 - 1
[[1,2,3],[4]]
=> 001 => 010 => 101 => 2 = 3 - 1
[[1,3],[2,4]]
=> 101 => 110 => 001 => 1 = 2 - 1
[[1,2],[3,4]]
=> 010 => 100 => 011 => 2 = 3 - 1
[[1,4],[2],[3]]
=> 110 => 011 => 100 => 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => 110 => 001 => 1 = 2 - 1
[[1,2],[3],[4]]
=> 011 => 101 => 010 => 1 = 2 - 1
[[1],[2],[3],[4]]
=> 111 => 111 => 000 => 0 = 1 - 1
[[1,2,3,4,5]]
=> 0000 => 0000 => 1111 => 4 = 5 - 1
[[1,3,4,5],[2]]
=> 1000 => 0001 => 1110 => 3 = 4 - 1
[[1,2,4,5],[3]]
=> 0100 => 1000 => 0111 => 3 = 4 - 1
[[1,2,3,5],[4]]
=> 0010 => 0100 => 1011 => 3 = 4 - 1
[[1,2,3,4],[5]]
=> 0001 => 0010 => 1101 => 3 = 4 - 1
[[1,3,5],[2,4]]
=> 1010 => 1100 => 0011 => 2 = 3 - 1
[[1,2,5],[3,4]]
=> 0100 => 1000 => 0111 => 3 = 4 - 1
[[1,3,4],[2,5]]
=> 1001 => 0110 => 1001 => 2 = 3 - 1
[[1,2,4],[3,5]]
=> 0101 => 1010 => 0101 => 2 = 3 - 1
[[1,2,3],[4,5]]
=> 0010 => 0100 => 1011 => 3 = 4 - 1
[[1,4,5],[2],[3]]
=> 1100 => 0011 => 1100 => 2 = 3 - 1
[[1,3,5],[2],[4]]
=> 1010 => 1100 => 0011 => 2 = 3 - 1
[[1,2,5],[3],[4]]
=> 0110 => 1001 => 0110 => 2 = 3 - 1
[[1,3,4],[2],[5]]
=> 1001 => 0110 => 1001 => 2 = 3 - 1
[[1,2,4],[3],[5]]
=> 0101 => 1010 => 0101 => 2 = 3 - 1
[[1,2,3],[4],[5]]
=> 0011 => 0101 => 1010 => 2 = 3 - 1
[[1,4],[2,5],[3]]
=> 1101 => 1110 => 0001 => 1 = 2 - 1
[[1,3],[2,5],[4]]
=> 1010 => 1100 => 0011 => 2 = 3 - 1
[[1,2],[3,5],[4]]
=> 0110 => 1001 => 0110 => 2 = 3 - 1
[[1,3],[2,4],[5]]
=> 1011 => 1101 => 0010 => 1 = 2 - 1
[[1,2],[3,4],[5]]
=> 0101 => 1010 => 0101 => 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> 1110 => 0111 => 1000 => 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => 1110 => 0001 => 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> 1011 => 1101 => 0010 => 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> 0111 => 1011 => 0100 => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => 1111 => 0000 => 0 = 1 - 1
[[1,2,3,4,5,6]]
=> 00000 => 00000 => 11111 => 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> 10000 => 00001 => 11110 => 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> 01000 => 10000 => 01111 => 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> 00100 => 01000 => 10111 => 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> 00010 => 00100 => 11011 => 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> 00001 => 00010 => 11101 => 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> 10100 => 11000 => 00111 => 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> 01000 => 10000 => 01111 => 4 = 5 - 1
[]
=> => => => ? = 1 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000394
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 0 = 1 - 1
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,4,6,7],[5,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6,8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,7],[4,6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,6,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,6,7],[4,8],[5]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,8],[6]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,8],[6]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,6],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6],[4,7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,6],[5,7],[8]]
=> 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,6,7],[4],[5],[8]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4],[6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5],[6],[8]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,6],[4],[7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,6],[5],[7],[8]]
=> 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,7],[4,6,8],[5]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,7],[4,5,8],[6]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6],[4,5,7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,5],[4,6,7],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4],[5,6,7],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,7],[4,6],[5,8]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,7],[4,5],[6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6],[4,7],[5,8]]
=> 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[1,2,3,5],[4,7],[6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4],[5,7],[6,8]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,7],[4,8],[5],[6]]
=> 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[1,2,3,7],[4,6],[5],[8]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,7],[4,5],[6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6],[4,7],[5],[8]]
=> 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[1,2,3,5],[4,7],[6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4],[5,7],[6],[8]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6],[4,5],[7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,5],[4,6],[7],[8]]
=> 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[[1,2,3,4],[5,6],[7],[8]]
=> 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,7],[4],[5],[6],[8]]
=> 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[1,2,3,6],[4],[5],[7],[8]]
=> 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[1,2,3,5],[4],[6],[7],[8]]
=> 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[[1,2,3],[4,6,7],[5,8]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3],[4,5,7],[6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3],[4,6,7],[5],[8]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3],[4,5,7],[6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3],[4,5,6],[7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3],[4,7],[5,8],[6]]
=> 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[1,2,3],[4,6],[5,7],[8]]
=> 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[1,2,3],[4,5],[6,7],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3],[4,7],[5],[6],[8]]
=> 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000676
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 5
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 4
[[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3,8]]
=> [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4,8]]
=> [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6
[[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3],[8]]
=> [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4],[8]]
=> [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6],[8]]
=> [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,3,5,6,7],[2,4,8]]
=> [2,4,8,1,3,5,6,7] => [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,4,8]]
=> [3,4,8,1,2,5,6,7] => [[.,[.,.]],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[[1,3,4,6,7],[2,5,8]]
=> [2,5,8,1,3,4,6,7] => [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3,5,8]]
=> [3,5,8,1,2,4,6,7] => ?
=> ?
=> ? = 5
[[1,2,3,6,7],[4,5,8]]
=> [4,5,8,1,2,3,6,7] => [[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,3,4,5,7],[2,6,8]]
=> [2,6,8,1,3,4,5,7] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 5
[[1,2,4,5,7],[3,6,8]]
=> [3,6,8,1,2,4,5,7] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 5
[[1,2,3,5,7],[4,6,8]]
=> [4,6,8,1,2,3,5,7] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 5
[[1,2,3,4,7],[5,6,8]]
=> [5,6,8,1,2,3,4,7] => [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 6
[[1,4,5,6,7],[2,8],[3]]
=> [3,2,8,1,4,5,6,7] => [[[.,.],.],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2,8],[4]]
=> [4,2,8,1,3,5,6,7] => [[[.,.],[.,.]],[[.,[.,[.,.]]],.]]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 5
[[1,2,5,6,7],[3,8],[4]]
=> [4,3,8,1,2,5,6,7] => ?
=> ?
=> ? = 5
[[1,3,4,6,7],[2,8],[5]]
=> [5,2,8,1,3,4,6,7] => ?
=> ?
=> ? = 5
[[1,2,4,6,7],[3,8],[5]]
=> [5,3,8,1,2,4,6,7] => ?
=> ?
=> ? = 5
[[1,2,3,6,7],[4,8],[5]]
=> [5,4,8,1,2,3,6,7] => [[[.,[.,[.,.]]],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[1,3,4,5,7],[2,8],[6]]
=> [6,2,8,1,3,4,5,7] => [[[.,.],[.,[.,[.,.]]]],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,4,5,7],[3,8],[6]]
=> [6,3,8,1,2,4,5,7] => [[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,5,7],[4,8],[6]]
=> [6,4,8,1,2,3,5,7] => [[[.,[.,[.,.]]],[.,.]],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 5
[[1,2,3,4,7],[5,8],[6]]
=> [6,5,8,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],.],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5
[[1,3,5,6,7],[2,4],[8]]
=> [8,2,4,1,3,5,6,7] => [[[.,.],[[.,.],[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,4],[8]]
=> [8,3,4,1,2,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,4,6,7],[2,5],[8]]
=> [8,2,5,1,3,4,6,7] => ?
=> ?
=> ? = 5
[[1,2,4,6,7],[3,5],[8]]
=> [8,3,5,1,2,4,6,7] => [[[.,[.,.]],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,6,7],[4,5],[8]]
=> [8,4,5,1,2,3,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 6
[[1,3,4,5,7],[2,6],[8]]
=> [8,2,6,1,3,4,5,7] => ?
=> ?
=> ? = 5
[[1,2,4,5,7],[3,6],[8]]
=> [8,3,6,1,2,4,5,7] => [[[.,[.,.]],[[.,[.,.]],[.,.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 5
[[1,2,3,5,7],[4,6],[8]]
=> [8,4,6,1,2,3,5,7] => ?
=> ?
=> ? = 5
[[1,2,3,4,7],[5,6],[8]]
=> [8,5,6,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,6],[2,7],[8]]
=> [8,2,7,1,3,4,5,6] => ?
=> ?
=> ? = 5
[[1,2,4,5,6],[3,7],[8]]
=> [8,3,7,1,2,4,5,6] => [[[.,[.,.]],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 5
[[1,2,3,5,6],[4,7],[8]]
=> [8,4,7,1,2,3,5,6] => [[[.,[.,[.,.]]],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 5
[[1,2,3,4,6],[5,7],[8]]
=> [8,5,7,1,2,3,4,6] => [[[.,[.,[.,[.,.]]]],[[.,.],.]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5
[[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,4,5,6,7],[2],[3],[8]]
=> [8,3,2,1,4,5,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2],[4],[8]]
=> [8,4,2,1,3,5,6,7] => [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 5
[[1,2,5,6,7],[3],[4],[8]]
=> [8,4,3,1,2,5,6,7] => ?
=> ?
=> ? = 5
[[1,3,4,6,7],[2],[5],[8]]
=> [8,5,2,1,3,4,6,7] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3],[5],[8]]
=> [8,5,3,1,2,4,6,7] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 5
[[1,2,3,6,7],[4],[5],[8]]
=> [8,5,4,1,2,3,6,7] => [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
[[1,3,4,5,7],[2],[6],[8]]
=> [8,6,2,1,3,4,5,7] => ?
=> ?
=> ? = 5
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 1
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 2
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 3
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 5
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 5
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,2,3,4,5,6,7],[8]]
=> 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,3,4,5,6,7],[2,8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6,8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,3,4,5,6,7],[2],[8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,6],[7],[8]]
=> 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[[1,3,5,6,7],[2,4,8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,4,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,6,7],[2,5,8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3,5,8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,6,7],[4,5,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,7],[2,6,8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,4,5,7],[3,6,8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,5,7],[4,6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,4,7],[5,6,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,4,5,6,7],[2,8],[3]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2,8],[4]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,8],[4]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,3,4,6,7],[2,8],[5]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3,8],[5]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,6,7],[4,8],[5]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,3,4,5,7],[2,8],[6]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,4,5,7],[3,8],[6]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,5,7],[4,8],[6]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,4,7],[5,8],[6]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2,4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,4],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,6,7],[2,5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3,5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,6,7],[4,5],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,7],[2,6],[8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,4,5,7],[3,6],[8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,5,7],[4,6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,4,7],[5,6],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,6],[2,7],[8]]
=> 1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,4,5,6],[3,7],[8]]
=> 0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,3,5,6],[4,7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,3,4,6],[5,7],[8]]
=> 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[[1,2,3,4,5],[6,7],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,4,5,6,7],[2],[3],[8]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2],[4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3],[4],[8]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,3,4,6,7],[2],[5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3],[5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001088
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 1
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 2
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 3
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 5
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 5
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,2,3,4,5,6,7],[8]]
=> 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,3,4,5,6,7],[2,8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6,8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,3,4,5,6,7],[2],[8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,6],[7],[8]]
=> 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[[1,3,5,6,7],[2,4,8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,4,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,6,7],[2,5,8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3,5,8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,6,7],[4,5,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,7],[2,6,8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,4,5,7],[3,6,8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,5,7],[4,6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,4,7],[5,6,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,4,5,6,7],[2,8],[3]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2,8],[4]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,8],[4]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,3,4,6,7],[2,8],[5]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3,8],[5]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,6,7],[4,8],[5]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,3,4,5,7],[2,8],[6]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,4,5,7],[3,8],[6]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,5,7],[4,8],[6]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,4,7],[5,8],[6]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2,4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3,4],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,6,7],[2,5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3,5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,3,6,7],[4,5],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,7],[2,6],[8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,4,5,7],[3,6],[8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,5,7],[4,6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,3,4,7],[5,6],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,6],[2,7],[8]]
=> 1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,4,5,6],[3,7],[8]]
=> 0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,3,5,6],[4,7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5
[[1,2,3,4,6],[5,7],[8]]
=> 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[[1,2,3,4,5],[6,7],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,4,5,6,7],[2],[3],[8]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5
[[1,3,5,6,7],[2],[4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,2,5,6,7],[3],[4],[8]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5
[[1,3,4,6,7],[2],[5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,4,6,7],[3],[5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St000024
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 0 = 1 - 1
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,4,5,6,7],[8]]
=> 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[[1,3,4,5,6,7],[2,8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6,8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,6,7],[2],[8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,6],[7],[8]]
=> 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,3,5,6,7],[2,4,8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3,4,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,6,7],[2,5,8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3,5,8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6,7],[4,5,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,7],[2,6,8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,7],[3,6,8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,6,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,4,5,6,7],[2,8],[3]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2,8],[4]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3,8],[4]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,4,6,7],[2,8],[5]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3,8],[5]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6,7],[4,8],[5]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,4,5,7],[2,8],[6]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,7],[3,8],[6]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,8],[6]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,8],[6]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2,4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3,4],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,6,7],[2,5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3,5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6,7],[4,5],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,7],[2,6],[8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,7],[3,6],[8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,6],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,6],[2,7],[8]]
=> 1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,6],[3,7],[8]]
=> 0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,6],[4,7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,6],[5,7],[8]]
=> 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,4,5,6,7],[2],[3],[8]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2],[4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3],[4],[8]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,4,6,7],[2],[5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3],[5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St001189
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 0 = 1 - 1
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,4,5,6,7],[8]]
=> 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[[1,3,4,5,6,7],[2,8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6,8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,6,7],[2],[8]]
=> 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,6],[7],[8]]
=> 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,3,5,6,7],[2,4,8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3,4,8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,6,7],[2,5,8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3,5,8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6,7],[4,5,8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,7],[2,6,8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,7],[3,6,8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6,8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,6,8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,4,5,6,7],[2,8],[3]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2,8],[4]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3,8],[4]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,4,6,7],[2,8],[5]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3,8],[5]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6,7],[4,8],[5]]
=> 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,4,5,7],[2,8],[6]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,7],[3,8],[6]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,8],[6]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,8],[6]]
=> 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2,4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3,4],[8]]
=> 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,6,7],[2,5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3,5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,6,7],[4,5],[8]]
=> 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,7],[2,6],[8]]
=> 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,7],[3,6],[8]]
=> 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,7],[4,6],[8]]
=> 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,7],[5,6],[8]]
=> 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,6],[2,7],[8]]
=> 1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,4,5,6],[3,7],[8]]
=> 0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,5,6],[4,7],[8]]
=> 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,6],[5,7],[8]]
=> 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 1
[[1,2,3,4,5],[6,7],[8]]
=> 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,4,5,6,7],[2],[3],[8]]
=> 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,5,6,7],[2],[4],[8]]
=> 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,5,6,7],[3],[4],[8]]
=> 0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[1,3,4,6,7],[2],[5],[8]]
=> 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[[1,2,4,6,7],[3],[5],[8]]
=> 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000632The jump number of the poset. St000245The number of ascents of a permutation. St000167The number of leaves of an ordered tree. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000021The number of descents of a permutation. 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. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000702The number of weak deficiencies of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St000155The number of exceedances (also excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St001726The number of visible inversions of a permutation. St000039The number of crossings of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001896The number of right descents of a signed permutations.
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