Your data matches 307 different statistics following compositions of up to 3 maps.
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Mp00071: Permutations descent compositionInteger compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1] => 1
[1,3,2] => [2,1] => 2
[2,1,3] => [1,2] => 1
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [1,1,1] => 3
[1,2,4,3] => [3,1] => 3
[1,3,2,4] => [2,2] => 2
[1,3,4,2] => [3,1] => 3
[1,4,2,3] => [2,2] => 2
[2,1,3,4] => [1,3] => 1
[2,3,1,4] => [2,2] => 2
[2,3,4,1] => [3,1] => 3
[2,4,1,3] => [2,2] => 2
[3,1,2,4] => [1,3] => 1
[3,2,1,4] => [1,1,2] => 3
[3,4,1,2] => [2,2] => 2
[4,1,2,3] => [1,3] => 1
[4,2,1,3] => [1,1,2] => 3
[4,3,1,2] => [1,1,2] => 3
[1,2,3,5,4] => [4,1] => 4
[1,2,4,3,5] => [3,2] => 3
[1,2,4,5,3] => [4,1] => 4
[1,2,5,3,4] => [3,2] => 3
[1,3,2,4,5] => [2,3] => 2
[1,3,4,2,5] => [3,2] => 3
[1,3,4,5,2] => [4,1] => 4
[1,3,5,2,4] => [3,2] => 3
[1,4,2,3,5] => [2,3] => 2
[1,4,5,2,3] => [3,2] => 3
[1,5,2,3,4] => [2,3] => 2
[2,1,3,4,5] => [1,4] => 1
[2,3,1,4,5] => [2,3] => 2
[2,3,4,1,5] => [3,2] => 3
[2,3,4,5,1] => [4,1] => 4
[2,3,5,1,4] => [3,2] => 3
[2,4,1,3,5] => [2,3] => 2
[2,4,5,1,3] => [3,2] => 3
[2,5,1,3,4] => [2,3] => 2
[3,1,2,4,5] => [1,4] => 1
[3,2,1,4,5] => [1,1,3] => 3
[3,4,1,2,5] => [2,3] => 2
[3,4,5,1,2] => [3,2] => 3
[3,5,1,2,4] => [2,3] => 2
[4,1,2,3,5] => [1,4] => 1
[4,2,1,3,5] => [1,1,3] => 3
[4,3,1,2,5] => [1,1,3] => 3
[4,5,1,2,3] => [2,3] => 2
[5,1,2,3,4] => [1,4] => 1
[5,2,1,3,4] => [1,1,3] => 3
Description
The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [[1],[2]]
=> 1
[1,3,2] => [[1,2],[3]]
=> 2
[2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [[1,2],[3]]
=> 2
[3,1,2] => [[1,3],[2]]
=> 1
[3,2,1] => [[1],[2],[3]]
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,3,2,4] => [[1,2,4],[3]]
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,4,2,3] => [[1,2,4],[3]]
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,3,1,4] => [[1,2,4],[3]]
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> 3
[2,4,1,3] => [[1,2],[3,4]]
=> 2
[3,1,2,4] => [[1,3,4],[2]]
=> 1
[3,2,1,4] => [[1,4],[2],[3]]
=> 3
[3,4,1,2] => [[1,2],[3,4]]
=> 2
[4,1,2,3] => [[1,3,4],[2]]
=> 1
[4,2,1,3] => [[1,4],[2],[3]]
=> 3
[4,3,1,2] => [[1,4],[2],[3]]
=> 3
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> 3
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> 3
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> 3
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> 2
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> 3
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> 3
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> 2
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> 3
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> 2
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> 3
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> 3
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 3
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> 3
Description
The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Mp00109: Permutations descent wordBinary words
St000391: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => 1 => 1
[1,3,2] => 01 => 2
[2,1,3] => 10 => 1
[2,3,1] => 01 => 2
[3,1,2] => 10 => 1
[3,2,1] => 11 => 3
[1,2,4,3] => 001 => 3
[1,3,2,4] => 010 => 2
[1,3,4,2] => 001 => 3
[1,4,2,3] => 010 => 2
[2,1,3,4] => 100 => 1
[2,3,1,4] => 010 => 2
[2,3,4,1] => 001 => 3
[2,4,1,3] => 010 => 2
[3,1,2,4] => 100 => 1
[3,2,1,4] => 110 => 3
[3,4,1,2] => 010 => 2
[4,1,2,3] => 100 => 1
[4,2,1,3] => 110 => 3
[4,3,1,2] => 110 => 3
[1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => 0001 => 4
[1,2,5,3,4] => 0010 => 3
[1,3,2,4,5] => 0100 => 2
[1,3,4,2,5] => 0010 => 3
[1,3,4,5,2] => 0001 => 4
[1,3,5,2,4] => 0010 => 3
[1,4,2,3,5] => 0100 => 2
[1,4,5,2,3] => 0010 => 3
[1,5,2,3,4] => 0100 => 2
[2,1,3,4,5] => 1000 => 1
[2,3,1,4,5] => 0100 => 2
[2,3,4,1,5] => 0010 => 3
[2,3,4,5,1] => 0001 => 4
[2,3,5,1,4] => 0010 => 3
[2,4,1,3,5] => 0100 => 2
[2,4,5,1,3] => 0010 => 3
[2,5,1,3,4] => 0100 => 2
[3,1,2,4,5] => 1000 => 1
[3,2,1,4,5] => 1100 => 3
[3,4,1,2,5] => 0100 => 2
[3,4,5,1,2] => 0010 => 3
[3,5,1,2,4] => 0100 => 2
[4,1,2,3,5] => 1000 => 1
[4,2,1,3,5] => 1100 => 3
[4,3,1,2,5] => 1100 => 3
[4,5,1,2,3] => 0100 => 2
[5,1,2,3,4] => 1000 => 1
[5,2,1,3,4] => 1100 => 3
Description
The sum of the positions of the ones in a binary word.
Mp00064: Permutations reversePermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [1,2] => [[1,2]]
=> 1
[1,3,2] => [2,3,1] => [[1,2],[3]]
=> 2
[2,1,3] => [3,1,2] => [[1,3],[2]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> 2
[3,1,2] => [2,1,3] => [[1,3],[2]]
=> 1
[3,2,1] => [1,2,3] => [[1,2,3]]
=> 3
[1,2,4,3] => [3,4,2,1] => [[1,2],[3],[4]]
=> 3
[1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [2,4,3,1] => [[1,2],[3],[4]]
=> 3
[1,4,2,3] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[2,1,3,4] => [4,3,1,2] => [[1,4],[2],[3]]
=> 1
[2,3,1,4] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[2,3,4,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[2,4,1,3] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,1,2,4] => [4,2,1,3] => [[1,4],[2],[3]]
=> 1
[3,2,1,4] => [4,1,2,3] => [[1,3,4],[2]]
=> 3
[3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[4,1,2,3] => [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[4,2,1,3] => [3,1,2,4] => [[1,3,4],[2]]
=> 3
[4,3,1,2] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[1,2,3,5,4] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 4
[1,2,4,3,5] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 3
[1,2,4,5,3] => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 4
[1,2,5,3,4] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,3,4,5,2] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 4
[1,3,5,2,4] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 3
[1,5,2,3,4] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 2
[2,1,3,4,5] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 1
[2,3,1,4,5] => [5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> 2
[2,3,4,1,5] => [5,1,4,3,2] => [[1,3],[2],[4],[5]]
=> 3
[2,3,4,5,1] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[2,3,5,1,4] => [4,1,5,3,2] => [[1,3],[2,4],[5]]
=> 3
[2,4,1,3,5] => [5,3,1,4,2] => [[1,4],[2,5],[3]]
=> 2
[2,4,5,1,3] => [3,1,5,4,2] => [[1,3],[2,4],[5]]
=> 3
[2,5,1,3,4] => [4,3,1,5,2] => [[1,4],[2,5],[3]]
=> 2
[3,1,2,4,5] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 1
[3,2,1,4,5] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[3,4,1,2,5] => [5,2,1,4,3] => [[1,4],[2,5],[3]]
=> 2
[3,4,5,1,2] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[3,5,1,2,4] => [4,2,1,5,3] => [[1,4],[2,5],[3]]
=> 2
[4,1,2,3,5] => [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 1
[4,2,1,3,5] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 3
[4,3,1,2,5] => [5,2,1,3,4] => [[1,4,5],[2],[3]]
=> 3
[4,5,1,2,3] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
[5,1,2,3,4] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1
[5,2,1,3,4] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 3
Description
The charge of a standard tableau.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000081: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> 1
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
Description
The number of edges of a graph.
Matching statistic: St000169
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00085: Standard tableaux Schützenberger involutionStandard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [[1],[2]]
=> [[1],[2]]
=> 1
[1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[3,1,2] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3,2,1] => [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[1,4,2,3] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[2,3,1,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[2,4,1,3] => [[1,2],[3,4]]
=> [[1,2],[3,4]]
=> 2
[3,1,2,4] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 3
[3,4,1,2] => [[1,2],[3,4]]
=> [[1,2],[3,4]]
=> 2
[4,1,2,3] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 3
[4,3,1,2] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 3
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 3
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 3
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 3
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 2
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 3
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 2
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 3
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
Description
The cocharge of a standard tableau. The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm: 1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$. 2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000271: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> 1
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
Description
The chromatic index of a graph. This is the minimal number of colours needed such that no two adjacent edges have the same colour.
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000492: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [[1],[2]]
=> {{1},{2}}
=> 1
[1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 3
[3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,1,3] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 3
[4,3,1,2] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 3
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
Description
The rob statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rob''' (right-opener-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal element of a block.
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000499: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [[1],[2]]
=> {{1},{2}}
=> 1
[1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 3
[3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,1,3] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 3
[4,3,1,2] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 3
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 3
Description
The rcb statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcb''' (right-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[4,1,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[4,3,1,2,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[4,5,1,2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[5,1,2,3,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[5,2,1,3,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
Description
The major index east count of a Dyck path. The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]]. The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
The following 297 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001118The acyclic chromatic index of a graph. St001161The major index north count of a Dyck path. St001697The shifted natural comajor index of a standard Young tableau. St001721The degree of a binary word. St000468The Hosoya index of a graph. St000915The Ore degree of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St000012The area of a Dyck path. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000184The size of the centralizer of any permutation of given cycle type. St000185The weighted size of a partition. St000228The size of a partition. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000722The number of different neighbourhoods in a graph. St000784The maximum of the length and the largest part of the integer partition. St000822The Hadwiger number of the graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000926The clique-coclique number of a graph. St000984The number of boxes below precisely one peak. St000992The alternating sum of the parts of an integer partition. St001029The size of the core of a graph. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001249Sum of the odd parts of a partition. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001622The number of join-irreducible elements of a lattice. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001759The Rajchgot index of a permutation. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000145The Dyson rank of a partition. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000532The total number of rook placements on a Ferrers board. St000536The pathwidth of a graph. St000693The modular (standard) major index of a standard tableau. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000446The disorder of a permutation. St001812The biclique partition number of a graph. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000794The mak of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St001671Haglund's hag of a permutation. St000462The major index minus the number of excedences of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001379The number of inversions plus the major index of a permutation. St000795The mad of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001651The Frankl number of a lattice. St000833The comajor index of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000018The number of inversions of a permutation. St001875The number of simple modules with projective dimension at most 1. St001397Number of pairs of incomparable elements in a finite poset. St000740The last entry of a permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000246The number of non-inversions of a permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000004The major index of a permutation. St000305The inverse major index of a permutation. St000005The bounce statistic of a Dyck path. St000304The load of a permutation. St000796The stat' of a permutation. St001117The game chromatic index of a graph. St001428The number of B-inversions of a signed permutation. St000086The number of subgraphs. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000224The sorting index of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001621The number of atoms of a lattice. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St000450The number of edges minus the number of vertices plus 2 of a graph. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000259The diameter of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001568The smallest positive integer that does not appear twice in the partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001862The number of crossings of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001931The weak major index of an integer composition regarded as a word. St001570The minimal number of edges to add to make a graph Hamiltonian. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001209The pmaj statistic of a parking function. St001433The flag major index of a signed permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.