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Your data matches 146 different statistics following compositions of up to 3 maps.
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Matching statistic: St000157
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St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,2]]
=> 0
[[1],[2]]
=> 1
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 2
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 1
[[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 1
[[1,4],[2],[3]]
=> 2
[[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> 2
[[1],[2],[3],[4]]
=> 3
[[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> 1
[[1,2,4,5],[3]]
=> 1
[[1,2,3,5],[4]]
=> 1
[[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 1
[[1,3,4],[2,5]]
=> 2
[[1,2,4],[3,5]]
=> 2
[[1,2,3],[4,5]]
=> 1
[[1,4,5],[2],[3]]
=> 2
[[1,3,5],[2],[4]]
=> 2
[[1,2,5],[3],[4]]
=> 2
[[1,3,4],[2],[5]]
=> 2
[[1,2,4],[3],[5]]
=> 2
[[1,2,3],[4],[5]]
=> 2
[[1,4],[2,5],[3]]
=> 3
[[1,3],[2,5],[4]]
=> 2
[[1,2],[3,5],[4]]
=> 2
[[1,3],[2,4],[5]]
=> 3
[[1,2],[3,4],[5]]
=> 2
[[1,5],[2],[3],[4]]
=> 3
[[1,4],[2],[3],[5]]
=> 3
[[1,3],[2],[4],[5]]
=> 3
[[1,2],[3],[4],[5]]
=> 3
[[1],[2],[3],[4],[5]]
=> 4
[[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> 1
[[1,2,4,5,6],[3]]
=> 1
[[1,2,3,5,6],[4]]
=> 1
[[1,2,3,4,6],[5]]
=> 1
[[1,2,3,4,5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> 2
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: St000507
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(load all 3 compositions to match this statistic)
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1 = 0 + 1
[[1,2]]
=> 2 = 1 + 1
[[1],[2]]
=> 1 = 0 + 1
[[1,2,3]]
=> 3 = 2 + 1
[[1,3],[2]]
=> 2 = 1 + 1
[[1,2],[3]]
=> 2 = 1 + 1
[[1],[2],[3]]
=> 1 = 0 + 1
[[1,2,3,4]]
=> 4 = 3 + 1
[[1,3,4],[2]]
=> 3 = 2 + 1
[[1,2,4],[3]]
=> 3 = 2 + 1
[[1,2,3],[4]]
=> 3 = 2 + 1
[[1,3],[2,4]]
=> 2 = 1 + 1
[[1,2],[3,4]]
=> 3 = 2 + 1
[[1,4],[2],[3]]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> 1 = 0 + 1
[[1,2,3,4,5]]
=> 5 = 4 + 1
[[1,3,4,5],[2]]
=> 4 = 3 + 1
[[1,2,4,5],[3]]
=> 4 = 3 + 1
[[1,2,3,5],[4]]
=> 4 = 3 + 1
[[1,2,3,4],[5]]
=> 4 = 3 + 1
[[1,3,5],[2,4]]
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> 4 = 3 + 1
[[1,3,4],[2,5]]
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> 4 = 3 + 1
[[1,4,5],[2],[3]]
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 2 = 1 + 1
[[1,3],[2,5],[4]]
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[[1,2,3,4,5,6]]
=> 6 = 5 + 1
[[1,3,4,5,6],[2]]
=> 5 = 4 + 1
[[1,2,4,5,6],[3]]
=> 5 = 4 + 1
[[1,2,3,5,6],[4]]
=> 5 = 4 + 1
[[1,2,3,4,6],[5]]
=> 5 = 4 + 1
[[1,2,3,4,5],[6]]
=> 5 = 4 + 1
[[1,3,5,6],[2,4]]
=> 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: St001298
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(load all 5 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 0
[[1,2,3]]
=> [1,2,3] => 2
[[1,3],[2]]
=> [2,1,3] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => 0
[[1,2,3,4]]
=> [1,2,3,4] => 3
[[1,3,4],[2]]
=> [2,1,3,4] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => 2
[[1,2,3],[4]]
=> [4,1,2,3] => 2
[[1,3],[2,4]]
=> [2,4,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => 3
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 5
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 4
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 4
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 4
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 4
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 3
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St001489
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(load all 8 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 1
[[1,2,3]]
=> [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,2,3,4]]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => 1
[[1,2,4],[3]]
=> [3,1,2,4] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 3
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 3
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 2
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St000021
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(load all 13 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0
[[1,2]]
=> [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 3
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => 3
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 3
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => 3
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000245
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0
[[1,2]]
=> [1,2] => [1,2] => 1
[[1],[2]]
=> [2,1] => [2,1] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 2
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 3
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 2
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 2
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 4
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 3
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 3
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 3
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 4
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => 4
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => 4
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 4
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => 3
Description
The number of ascents of a permutation.
Matching statistic: St000325
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 1 = 0 + 1
[[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 2 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 3 = 2 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 3 = 2 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => 2 = 1 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 2 = 1 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 3 = 2 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => 3 = 2 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 3 = 2 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 3 = 2 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => 3 = 2 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 3 = 2 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 3 = 2 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 3 = 2 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => 4 = 3 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => 3 = 2 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => 4 = 3 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 5 = 4 + 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => 3 = 2 + 1
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 1 = 0 + 1
[[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 2 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 3 = 2 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 3 = 2 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => 2 = 1 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 2 = 1 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 3 = 2 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => 3 = 2 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 3 = 2 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 3 = 2 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => 3 = 2 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 3 = 2 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 3 = 2 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 3 = 2 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => 4 = 3 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => 3 = 2 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => 4 = 3 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 5 = 4 + 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => 3 = 2 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St001116
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001116: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001116: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> [2] => ([],2)
=> 1 = 0 + 1
[[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3]]
=> [3] => ([],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,3],[2],[4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> [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)
=> 5 = 4 + 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The game chromatic number of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic number is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St000024
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(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 0
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 3
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
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.
The following 136 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000053The number of valleys of the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000536The pathwidth of a graph. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000288The number of ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000354The number of recoils of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. 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. St001372The length of a longest cyclic run of ones of a binary word. St001480The number of simple summands of the module J^2/J^3. St000702The number of weak deficiencies of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001812The biclique partition number of a graph. St001331The size of the minimal feedback vertex set. St001321The number of vertices of the largest induced subforest of a graph. St001427The number of descents of a signed permutation. St001644The dimension of a graph. St000159The number of distinct parts of the integer partition. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St000731The number of double exceedences of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001330The hat guessing number of a graph. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001727The number of invisible inversions of a permutation. St000991The number of right-to-left minima of a permutation. St001726The number of visible inversions 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 St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. 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. St001060The distinguishing index of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St001896The number of right descents of a signed permutations. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001720The minimal length of a chain of small intervals in a lattice. St000455The second largest eigenvalue of a graph if it is integral. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001712The number of natural descents of a standard Young tableau. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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