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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St001712
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St001712: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0
[[1],[2]]
=> 0
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 0
[[1],[2],[3]]
=> 0
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 1
[[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> 0
[[1,3],[2,4]]
=> 1
[[1,2],[3,4]]
=> 0
[[1,4],[2],[3]]
=> 1
[[1,3],[2],[4]]
=> 1
[[1,2],[3],[4]]
=> 0
[[1],[2],[3],[4]]
=> 0
[[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]]
=> 0
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 1
[[1,3,4],[2,5]]
=> 1
[[1,2,4],[3,5]]
=> 1
[[1,2,3],[4,5]]
=> 0
[[1,4,5],[2],[3]]
=> 1
[[1,3,5],[2],[4]]
=> 2
[[1,2,5],[3],[4]]
=> 1
[[1,3,4],[2],[5]]
=> 1
[[1,2,4],[3],[5]]
=> 1
[[1,2,3],[4],[5]]
=> 0
[[1,4],[2,5],[3]]
=> 1
[[1,3],[2,5],[4]]
=> 2
[[1,2],[3,5],[4]]
=> 1
[[1,3],[2,4],[5]]
=> 1
[[1,2],[3,4],[5]]
=> 0
[[1,5],[2],[3],[4]]
=> 1
[[1,4],[2],[3],[5]]
=> 1
[[1,3],[2],[4],[5]]
=> 1
[[1,2],[3],[4],[5]]
=> 0
[[1],[2],[3],[4],[5]]
=> 0
[[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]]
=> 0
[[1,3,5,6],[2,4]]
=> 2
[[1,2,5,6],[3,4]]
=> 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St000354
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => 1
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 1
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0
[[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] => [1,3,4,5,2] => 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[[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] => [1,3,4,5,6,2] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => 2
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 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: St000646
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => 0
[[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] => 0
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0
[[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] => 0
[[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] => 0
[[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] => 0
[[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] => 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] => 0
[[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] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1
[[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] => 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 0
[[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] => 1
[[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] => 0
[[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] => 0
[[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] => 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] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => 2
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => 1
Description
The number of big ascents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see [[St000441]].
Matching statistic: St001840
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00112: Set partitions —complement⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> 0
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 2
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 1
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 2
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 1
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 2
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 1
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 1
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[[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]]
=> {{1,3,4,5,6},{2}}
=> {{1,2,3,4,6},{5}}
=> 1
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,3,5,6},{4}}
=> 1
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,4,5,6},{3}}
=> 1
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,3,4,5,6},{2}}
=> 1
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,2,4,6},{3,5}}
=> 2
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,5,6},{3,4}}
=> 1
Description
The number of descents of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The word $w$ has a descent at position $i$ if $w_i > w_{i+1}$.
Matching statistic: St000157
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ 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%
Mp00223: Permutations —runsort⟶ 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%
Values
[[1,2]]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[[1],[2]]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => [[1,2],[3]]
=> 1
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => [[1,2,3]]
=> 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [[1,2,3]]
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,2,3,4,5]]
=> [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] => [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,2,3,4,5,6]]
=> [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] => [1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => [[1,2,3,5,6],[4]]
=> 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => [[1,2,3,4,6],[5]]
=> 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
=> 2
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => [[1,2,3,4],[5,6]]
=> 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: St000647
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [2,1] => 0
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [2,3,1] => 1
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 1
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,3,2,1,4] => [4,3,2,5,1] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [3,5,2,4,1] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [3,2,5,4,1] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [3,5,2,1,4] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [3,2,5,1,4] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,1,2,4] => [3,4,2,5,1] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1,5,2,3] => [2,4,5,1,3] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,5,2,4] => [2,4,1,5,3] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,4,5,3] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,1,4,2,5] => [2,4,1,3,5] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,5,3,1,4,2] => [4,6,3,5,2,1] => 2
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [6,5,2,1,4,3] => [4,3,6,5,2,1] => 1
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St001489
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [1,2,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
=> [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] => [1,3,4,2] => 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [1,4,2,3] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,4,2,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [1,2,4,3] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [1,2,3,4] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [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] => [1,3,4,5,2] => 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [1,4,5,2,3] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [1,5,2,3,4] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => [1,4,2,5,3] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [1,2,5,3,4] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => [1,4,5,2,3] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => [1,5,2,3,4] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,4,5,2,3] => 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [1,5,2,4,3] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [1,5,2,3,4] => 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [1,2,4,5,3] => 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [1,2,5,3,4] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => [1,5,2,3,4] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [1,3,2,5,4] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,3,2,4,5] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => [1,2,5,3,4] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,5,2,3,4] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,2,5,3,4] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,2,3,5,4] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [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] => [1,3,4,5,6,2] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => [1,4,5,6,2,3] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => [1,5,6,2,3,4] => 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => [1,6,2,3,4,5] => 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => [1,4,2,5,6,3] => 2
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => 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: St000470
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
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] => [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,2,3,4,5]]
=> [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] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 2 = 1 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 2 = 1 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => [1,3,5,2,4] => 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [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] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => [1,5,2,6,3,4] => 3 = 2 + 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => 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: St000619
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,2,3,4,5]]
=> [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] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 2 = 1 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 2 = 1 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => [1,3,5,2,4] => 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [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] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => [1,5,2,6,3,4] => 3 = 2 + 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => 2 = 1 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000779
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => 0
[[1],[2]]
=> [2,1] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [3,4,2,1] => 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [4,5,3,2,1] => 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,5,4,2,1] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,5,4,3,1] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,5,3,1] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,2,5,4,1] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [2,3,5,4,1] => 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,4,5,3,2] => 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,4,1,5,2] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,3,1,5,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,4,2,5,3] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,3,4,5,1] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [5,6,4,3,2,1] => 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [4,6,5,3,2,1] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [3,6,5,4,2,1] => 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,6,5,4,3,1] => 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [5,3,6,4,2,1] => 2
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => 1
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St001896The number of right descents of a signed permutations. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001435The number of missing boxes in the first row. St001487The number of inner corners of a skew partition. St001488The number of corners of a skew partition.
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