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Your data matches 159 different statistics following compositions of up to 3 maps.
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Matching statistic: St000354
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 0
[[2,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 1
[[1,3]]
=> [1,2] => 0
[[2,3]]
=> [1,2] => 0
[[3,3]]
=> [1,2] => 0
[[1],[3]]
=> [2,1] => 1
[[2],[3]]
=> [2,1] => 1
[[1,1,2]]
=> [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => 1
[[1,2],[2]]
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => 0
[[2,4]]
=> [1,2] => 0
[[3,4]]
=> [1,2] => 0
[[4,4]]
=> [1,2] => 0
[[1],[4]]
=> [2,1] => 1
[[2],[4]]
=> [2,1] => 1
[[3],[4]]
=> [2,1] => 1
[[1,1,3]]
=> [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1,3],[2]]
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => 1
[[2,3],[3]]
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => 1
[[1,5]]
=> [1,2] => 0
[[2,5]]
=> [1,2] => 0
[[3,5]]
=> [1,2] => 0
[[4,5]]
=> [1,2] => 0
[[5,5]]
=> [1,2] => 0
[[1],[5]]
=> [2,1] => 1
[[2],[5]]
=> [2,1] => 1
[[3],[5]]
=> [2,1] => 1
[[4],[5]]
=> [2,1] => 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St001489
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Mp00075: Semistandard 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
[[1,2]]
=> [1,2] => 0
[[2,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 1
[[1,3]]
=> [1,2] => 0
[[2,3]]
=> [1,2] => 0
[[3,3]]
=> [1,2] => 0
[[1],[3]]
=> [2,1] => 1
[[2],[3]]
=> [2,1] => 1
[[1,1,2]]
=> [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => 1
[[1,2],[2]]
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => 0
[[2,4]]
=> [1,2] => 0
[[3,4]]
=> [1,2] => 0
[[4,4]]
=> [1,2] => 0
[[1],[4]]
=> [2,1] => 1
[[2],[4]]
=> [2,1] => 1
[[3],[4]]
=> [2,1] => 1
[[1,1,3]]
=> [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => 1
[[1,2],[3]]
=> [3,1,2] => 1
[[1,3],[2]]
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => 1
[[2,3],[3]]
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => 1
[[1,5]]
=> [1,2] => 0
[[2,5]]
=> [1,2] => 0
[[3,5]]
=> [1,2] => 0
[[4,5]]
=> [1,2] => 0
[[5,5]]
=> [1,2] => 0
[[1],[5]]
=> [2,1] => 1
[[2],[5]]
=> [2,1] => 1
[[3],[5]]
=> [2,1] => 1
[[4],[5]]
=> [2,1] => 1
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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Mp00075: Semistandard 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
[[1,2]]
=> [1,2] => [1,2] => 0
[[2,2]]
=> [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => 1
[[1,3]]
=> [1,2] => [1,2] => 0
[[2,3]]
=> [1,2] => [1,2] => 0
[[3,3]]
=> [1,2] => [1,2] => 0
[[1],[3]]
=> [2,1] => [2,1] => 1
[[2],[3]]
=> [2,1] => [2,1] => 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => 0
[[1,2,2]]
=> [1,2,3] => [1,2,3] => 0
[[2,2,2]]
=> [1,2,3] => [1,2,3] => 0
[[1,1],[2]]
=> [3,1,2] => [2,3,1] => 1
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => 1
[[1,4]]
=> [1,2] => [1,2] => 0
[[2,4]]
=> [1,2] => [1,2] => 0
[[3,4]]
=> [1,2] => [1,2] => 0
[[4,4]]
=> [1,2] => [1,2] => 0
[[1],[4]]
=> [2,1] => [2,1] => 1
[[2],[4]]
=> [2,1] => [2,1] => 1
[[3],[4]]
=> [2,1] => [2,1] => 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[2,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[2,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[3,3,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,1],[3]]
=> [3,1,2] => [2,3,1] => 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => [2,3,1] => 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,1,1],[2]]
=> [4,1,2,3] => [2,3,4,1] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => [2,3,1,4] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[1,5]]
=> [1,2] => [1,2] => 0
[[2,5]]
=> [1,2] => [1,2] => 0
[[3,5]]
=> [1,2] => [1,2] => 0
[[4,5]]
=> [1,2] => [1,2] => 0
[[5,5]]
=> [1,2] => [1,2] => 0
[[1],[5]]
=> [2,1] => [2,1] => 1
[[2],[5]]
=> [2,1] => [2,1] => 1
[[3],[5]]
=> [2,1] => [2,1] => 1
[[4],[5]]
=> [2,1] => [2,1] => 1
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: St000157
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion 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] => [[1,2]]
=> 0
[[2,2]]
=> [1,2] => [[1,2]]
=> 0
[[1],[2]]
=> [2,1] => [[1],[2]]
=> 1
[[1,3]]
=> [1,2] => [[1,2]]
=> 0
[[2,3]]
=> [1,2] => [[1,2]]
=> 0
[[3,3]]
=> [1,2] => [[1,2]]
=> 0
[[1],[3]]
=> [2,1] => [[1],[2]]
=> 1
[[2],[3]]
=> [2,1] => [[1],[2]]
=> 1
[[1,1,2]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[1,2,2]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[2,2,2]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[1,1],[2]]
=> [3,1,2] => [[1,2],[3]]
=> 1
[[1,2],[2]]
=> [2,1,3] => [[1,3],[2]]
=> 1
[[1,4]]
=> [1,2] => [[1,2]]
=> 0
[[2,4]]
=> [1,2] => [[1,2]]
=> 0
[[3,4]]
=> [1,2] => [[1,2]]
=> 0
[[4,4]]
=> [1,2] => [[1,2]]
=> 0
[[1],[4]]
=> [2,1] => [[1],[2]]
=> 1
[[2],[4]]
=> [2,1] => [[1],[2]]
=> 1
[[3],[4]]
=> [2,1] => [[1],[2]]
=> 1
[[1,1,3]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[1,2,3]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[1,3,3]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[2,2,3]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[2,3,3]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[3,3,3]]
=> [1,2,3] => [[1,2,3]]
=> 0
[[1,1],[3]]
=> [3,1,2] => [[1,2],[3]]
=> 1
[[1,2],[3]]
=> [3,1,2] => [[1,2],[3]]
=> 1
[[1,3],[2]]
=> [2,1,3] => [[1,3],[2]]
=> 1
[[1,3],[3]]
=> [2,1,3] => [[1,3],[2]]
=> 1
[[2,2],[3]]
=> [3,1,2] => [[1,2],[3]]
=> 1
[[2,3],[3]]
=> [2,1,3] => [[1,3],[2]]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [[1],[2],[3]]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[[1,5]]
=> [1,2] => [[1,2]]
=> 0
[[2,5]]
=> [1,2] => [[1,2]]
=> 0
[[3,5]]
=> [1,2] => [[1,2]]
=> 0
[[4,5]]
=> [1,2] => [[1,2]]
=> 0
[[5,5]]
=> [1,2] => [[1,2]]
=> 0
[[1],[5]]
=> [2,1] => [[1],[2]]
=> 1
[[2],[5]]
=> [2,1] => [[1],[2]]
=> 1
[[3],[5]]
=> [2,1] => [[1],[2]]
=> 1
[[4],[5]]
=> [2,1] => [[1],[2]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St001971
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001971: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001971: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => ([],2)
=> 0
[[2,2]]
=> [1,2] => ([],2)
=> 0
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => ([],2)
=> 0
[[2,3]]
=> [1,2] => ([],2)
=> 0
[[3,3]]
=> [1,2] => ([],2)
=> 0
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => ([],3)
=> 0
[[1,2,2]]
=> [1,2,3] => ([],3)
=> 0
[[2,2,2]]
=> [1,2,3] => ([],3)
=> 0
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> 1
[[1,4]]
=> [1,2] => ([],2)
=> 0
[[2,4]]
=> [1,2] => ([],2)
=> 0
[[3,4]]
=> [1,2] => ([],2)
=> 0
[[4,4]]
=> [1,2] => ([],2)
=> 0
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> 1
[[1,1,3]]
=> [1,2,3] => ([],3)
=> 0
[[1,2,3]]
=> [1,2,3] => ([],3)
=> 0
[[1,3,3]]
=> [1,2,3] => ([],3)
=> 0
[[2,2,3]]
=> [1,2,3] => ([],3)
=> 0
[[2,3,3]]
=> [1,2,3] => ([],3)
=> 0
[[3,3,3]]
=> [1,2,3] => ([],3)
=> 0
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> 1
[[1,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[1,5]]
=> [1,2] => ([],2)
=> 0
[[2,5]]
=> [1,2] => ([],2)
=> 0
[[3,5]]
=> [1,2] => ([],2)
=> 0
[[4,5]]
=> [1,2] => ([],2)
=> 0
[[5,5]]
=> [1,2] => ([],2)
=> 0
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> 1
Description
The number of negative eigenvalues of the adjacency matrix of the graph.
Matching statistic: St000325
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Mp00075: Semistandard 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
[[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,3]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,3]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[3,3]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[3]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[3,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[4,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[4]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[2],[4]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[3],[4]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[1,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[3,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[4,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[5,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[5]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[2],[5]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[3],[5]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[4],[5]]
=> [2,1] => [2,1] => 2 = 1 + 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 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00075: Semistandard 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
[[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,3]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,3]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[3,3]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[3]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[3,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[4,4]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[4]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[2],[4]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[3],[4]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[1,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[2,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[3,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[4,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[5,5]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[5]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[2],[5]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[3],[5]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[4],[5]]
=> [2,1] => [2,1] => 2 = 1 + 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: St000024
Mp00075: Semistandard 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
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: 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
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ 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] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[2]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[[1,1],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000083
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000083: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000083: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[2,2]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[1],[2]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[1,3]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[2,3]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[3,3]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[1],[3]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[2],[3]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[1,1,2]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[1,2,2]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[2,2,2]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> 1
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> 1
[[1,4]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[2,4]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[3,4]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[4,4]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[1],[4]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[2],[4]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[3],[4]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[1,1,3]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[1,3,3]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[2,2,3]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[2,3,3]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[3,3,3]]
=> [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> 1
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> 1
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> 1
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> 1
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> 1
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[[1,5]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[2,5]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[3,5]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[4,5]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[5,5]]
=> [1,2] => [2,1] => [[.,.],.]
=> 0
[[1],[5]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[2],[5]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[3],[5]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
[[4],[5]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1
Description
The number of left oriented leafs of a binary tree except the first one.
In other other words, this is the sum of canopee vector of the tree.
The canopee of a non empty binary tree T with n internal nodes is the list l of 0 and 1 of length n-1 obtained by going along the leaves of T from left to right except the two extremal ones, writing 0 if the leaf is a right leaf and 1 if the leaf is a left leaf.
This is also the number of nodes having a right child. Indeed each of said right children will give exactly one left oriented leaf.
The following 149 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000245The number of ascents of a permutation. St000288The number of ones in a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000167The number of leaves of an ordered tree. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. 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. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001896The number of right descents of a signed permutations. St000260The radius of a connected graph. St000668The least common multiple of the parts of the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001946The number of descents in a parking function. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000567The sum of the products of all pairs of parts. 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. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the 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. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given 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. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. 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. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001609The number of coloured trees such that the multiplicities of colours 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. 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. 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. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001330The hat guessing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000928The sum of the coefficients of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. 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. St000934The 2-degree 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. St001128The exponens consonantiae of a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000379The number of Hamiltonian cycles in a graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000259The diameter of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. 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. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St000264The girth of a graph, which is not a tree. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.
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