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Your data matches 96 different statistics following compositions of up to 3 maps.
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Matching statistic: St000393
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(load all 20 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 1
[2,1] => 1 => 1
[1,2,3] => 00 => 2
[1,3,2] => 01 => 1
[2,1,3] => 10 => 2
[2,3,1] => 01 => 1
[3,1,2] => 10 => 2
[3,2,1] => 11 => 2
[1,2,3,4] => 000 => 3
[1,2,4,3] => 001 => 2
[1,3,2,4] => 010 => 2
[1,3,4,2] => 001 => 2
[1,4,2,3] => 010 => 2
[1,4,3,2] => 011 => 2
[2,1,3,4] => 100 => 3
[2,1,4,3] => 101 => 2
[2,3,1,4] => 010 => 2
[2,3,4,1] => 001 => 2
[2,4,1,3] => 010 => 2
[2,4,3,1] => 011 => 2
[3,1,2,4] => 100 => 3
[3,1,4,2] => 101 => 2
[3,2,1,4] => 110 => 3
[3,2,4,1] => 101 => 2
[3,4,1,2] => 010 => 2
[3,4,2,1] => 011 => 2
[4,1,2,3] => 100 => 3
[4,1,3,2] => 101 => 2
[4,2,1,3] => 110 => 3
[4,2,3,1] => 101 => 2
[4,3,1,2] => 110 => 3
[4,3,2,1] => 111 => 3
[1,2,3,4,5] => 0000 => 4
[1,2,3,5,4] => 0001 => 3
[1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => 0001 => 3
[1,2,5,3,4] => 0010 => 3
[1,2,5,4,3] => 0011 => 3
[1,3,2,4,5] => 0100 => 3
[1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => 0010 => 3
[1,3,4,5,2] => 0001 => 3
[1,3,5,2,4] => 0010 => 3
[1,3,5,4,2] => 0011 => 3
[1,4,2,3,5] => 0100 => 3
[1,4,2,5,3] => 0101 => 2
[1,4,3,2,5] => 0110 => 3
[1,4,3,5,2] => 0101 => 2
[1,4,5,2,3] => 0010 => 3
[1,4,5,3,2] => 0011 => 3
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[2,1] => [1,2] => [1,1]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[3,1,2] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> 4 = 3 + 1
Description
The length of the partition.
Matching statistic: St000384
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[2,1] => [1,2] => [1,1]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[3,1,2] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> 4 = 3 + 1
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000507
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[2,1] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[3,1,2] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[3,2,1] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,3,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 3 = 2 + 1
[2,1,3,4] => [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 3 = 2 + 1
[2,3,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[3,1,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 3 = 2 + 1
[3,2,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> 3 = 2 + 1
[3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 4 = 3 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000784
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[2,1] => [1,2] => [1,1]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[3,1,2] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 2 + 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 3 = 2 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 2 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> 4 = 3 + 1
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St000380
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> 3 = 1 + 2
[2,1] => [1,2] => [1,1]
=> 3 = 1 + 2
[1,2,3] => [1,2,3] => [1,1,1]
=> 4 = 2 + 2
[1,3,2] => [1,3,2] => [2,1]
=> 3 = 1 + 2
[2,1,3] => [1,3,2] => [2,1]
=> 3 = 1 + 2
[2,3,1] => [1,2,3] => [1,1,1]
=> 4 = 2 + 2
[3,1,2] => [1,2,3] => [1,1,1]
=> 4 = 2 + 2
[3,2,1] => [1,2,3] => [1,1,1]
=> 4 = 2 + 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 4 = 2 + 2
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 4 = 2 + 2
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 4 = 2 + 2
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 4 = 2 + 2
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 4 = 2 + 2
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 4 = 2 + 2
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 4 = 2 + 2
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 4 = 2 + 2
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 4 = 2 + 2
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 4 = 2 + 2
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 4 = 2 + 2
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 4 = 2 + 2
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 4 = 2 + 2
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 4 = 2 + 2
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 4 = 2 + 2
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 4 = 2 + 2
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 5 = 3 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 6 = 4 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 5 = 3 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 5 = 3 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 5 = 3 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 5 = 3 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 5 = 3 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 5 = 3 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 4 = 2 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 5 = 3 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 5 = 3 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 5 = 3 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 5 = 3 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 5 = 3 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 4 = 2 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 4 = 2 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 5 = 3 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 5 = 3 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> 5 = 3 + 2
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St000024
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
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: St000053
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
Description
The number of valleys of the Dyck path.
Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[2,1] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[2,1,3] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[2,3,1] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[3,1,2] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[3,2,1] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
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: St000288
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ 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%
Mp00131: Permutations —descent bottoms⟶ 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
[1,2] => [1,2] => 0 => 1 => 1
[2,1] => [1,2] => 0 => 1 => 1
[1,2,3] => [1,2,3] => 00 => 11 => 2
[1,3,2] => [1,3,2] => 01 => 10 => 1
[2,1,3] => [1,3,2] => 01 => 10 => 1
[2,3,1] => [1,2,3] => 00 => 11 => 2
[3,1,2] => [1,2,3] => 00 => 11 => 2
[3,2,1] => [1,2,3] => 00 => 11 => 2
[1,2,3,4] => [1,2,3,4] => 000 => 111 => 3
[1,2,4,3] => [1,2,4,3] => 001 => 110 => 2
[1,3,2,4] => [1,3,2,4] => 010 => 101 => 2
[1,3,4,2] => [1,3,4,2] => 010 => 101 => 2
[1,4,2,3] => [1,4,2,3] => 010 => 101 => 2
[1,4,3,2] => [1,4,2,3] => 010 => 101 => 2
[2,1,3,4] => [1,3,4,2] => 010 => 101 => 2
[2,1,4,3] => [1,4,2,3] => 010 => 101 => 2
[2,3,1,4] => [1,4,2,3] => 010 => 101 => 2
[2,3,4,1] => [1,2,3,4] => 000 => 111 => 3
[2,4,1,3] => [1,3,2,4] => 010 => 101 => 2
[2,4,3,1] => [1,2,4,3] => 001 => 110 => 2
[3,1,2,4] => [1,2,4,3] => 001 => 110 => 2
[3,1,4,2] => [1,4,2,3] => 010 => 101 => 2
[3,2,1,4] => [1,4,2,3] => 010 => 101 => 2
[3,2,4,1] => [1,2,4,3] => 001 => 110 => 2
[3,4,1,2] => [1,2,3,4] => 000 => 111 => 3
[3,4,2,1] => [1,2,3,4] => 000 => 111 => 3
[4,1,2,3] => [1,2,3,4] => 000 => 111 => 3
[4,1,3,2] => [1,3,2,4] => 010 => 101 => 2
[4,2,1,3] => [1,3,2,4] => 010 => 101 => 2
[4,2,3,1] => [1,2,3,4] => 000 => 111 => 3
[4,3,1,2] => [1,2,3,4] => 000 => 111 => 3
[4,3,2,1] => [1,2,3,4] => 000 => 111 => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 1111 => 4
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1110 => 3
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1101 => 3
[1,2,4,5,3] => [1,2,4,5,3] => 0010 => 1101 => 3
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1101 => 3
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1101 => 3
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1011 => 3
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 2
[1,3,4,2,5] => [1,3,4,2,5] => 0100 => 1011 => 3
[1,3,4,5,2] => [1,3,4,5,2] => 0100 => 1011 => 3
[1,3,5,2,4] => [1,3,5,2,4] => 0100 => 1011 => 3
[1,3,5,4,2] => [1,3,5,2,4] => 0100 => 1011 => 3
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1011 => 3
[1,4,2,5,3] => [1,4,2,5,3] => 0110 => 1001 => 2
[1,4,3,2,5] => [1,4,2,5,3] => 0110 => 1001 => 2
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1011 => 3
[1,4,5,2,3] => [1,4,5,2,3] => 0100 => 1011 => 3
[1,4,5,3,2] => [1,4,5,2,3] => 0100 => 1011 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001622The number of join-irreducible elements of a lattice. St000021The number of descents of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000245The number of ascents of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St000167The number of leaves of an ordered tree. St000863The length of the first row of the shifted shape of a permutation. St000702The number of weak deficiencies of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000354The number of recoils of a permutation. St000264The girth of a graph, which is not a tree. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000054The first entry of the permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000141The maximum drop size of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000619The number of cyclic descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000213The number of weak exceedances (also weak excedences) of a permutation. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000155The number of exceedances (also excedences) 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. St000829The Ulam distance of a permutation to the identity permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000317The cycle descent number of a 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001427The number of descents of a signed permutation. St001820The size of the image of the pop stack sorting operator. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001668The number of points of the poset minus the width of the poset. St001960The number of descents of a permutation minus one if its first entry is not one. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001863The number of weak excedances of a signed permutation. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. St001626The number of maximal proper sublattices of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St000077The number of boxed and circled entries. St001271The competition number of a graph.
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