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Your data matches 325 different statistics following compositions of up to 3 maps.
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Matching statistic: St000021
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000021: 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 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: St000541
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000541: 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 indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000010
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Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2]
=> 1 = 0 + 1
[[2,2]]
=> [2]
=> 1 = 0 + 1
[[1],[2]]
=> [1,1]
=> 2 = 1 + 1
[[1,3]]
=> [2]
=> 1 = 0 + 1
[[2,3]]
=> [2]
=> 1 = 0 + 1
[[3,3]]
=> [2]
=> 1 = 0 + 1
[[1],[3]]
=> [1,1]
=> 2 = 1 + 1
[[2],[3]]
=> [1,1]
=> 2 = 1 + 1
[[1,1,2]]
=> [3]
=> 1 = 0 + 1
[[1,2,2]]
=> [3]
=> 1 = 0 + 1
[[2,2,2]]
=> [3]
=> 1 = 0 + 1
[[1,1],[2]]
=> [2,1]
=> 2 = 1 + 1
[[1,2],[2]]
=> [2,1]
=> 2 = 1 + 1
[[1,4]]
=> [2]
=> 1 = 0 + 1
[[2,4]]
=> [2]
=> 1 = 0 + 1
[[3,4]]
=> [2]
=> 1 = 0 + 1
[[4,4]]
=> [2]
=> 1 = 0 + 1
[[1],[4]]
=> [1,1]
=> 2 = 1 + 1
[[2],[4]]
=> [1,1]
=> 2 = 1 + 1
[[3],[4]]
=> [1,1]
=> 2 = 1 + 1
[[1,1,3]]
=> [3]
=> 1 = 0 + 1
[[1,2,3]]
=> [3]
=> 1 = 0 + 1
[[1,3,3]]
=> [3]
=> 1 = 0 + 1
[[2,2,3]]
=> [3]
=> 1 = 0 + 1
[[2,3,3]]
=> [3]
=> 1 = 0 + 1
[[3,3,3]]
=> [3]
=> 1 = 0 + 1
[[1,1],[3]]
=> [2,1]
=> 2 = 1 + 1
[[1,2],[3]]
=> [2,1]
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1]
=> 2 = 1 + 1
[[1,3],[3]]
=> [2,1]
=> 2 = 1 + 1
[[2,2],[3]]
=> [2,1]
=> 2 = 1 + 1
[[2,3],[3]]
=> [2,1]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 3 = 2 + 1
[[1,1,1,2]]
=> [4]
=> 1 = 0 + 1
[[1,1,2,2]]
=> [4]
=> 1 = 0 + 1
[[1,2,2,2]]
=> [4]
=> 1 = 0 + 1
[[2,2,2,2]]
=> [4]
=> 1 = 0 + 1
[[1,1,1],[2]]
=> [3,1]
=> 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1]
=> 2 = 1 + 1
[[1,2,2],[2]]
=> [3,1]
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [2,2]
=> 2 = 1 + 1
[[1,5]]
=> [2]
=> 1 = 0 + 1
[[2,5]]
=> [2]
=> 1 = 0 + 1
[[3,5]]
=> [2]
=> 1 = 0 + 1
[[4,5]]
=> [2]
=> 1 = 0 + 1
[[5,5]]
=> [2]
=> 1 = 0 + 1
[[1],[5]]
=> [1,1]
=> 2 = 1 + 1
[[2],[5]]
=> [1,1]
=> 2 = 1 + 1
[[3],[5]]
=> [1,1]
=> 2 = 1 + 1
[[4],[5]]
=> [1,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000325
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1 = 0 + 1
[[2,2]]
=> [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => 2 = 1 + 1
[[1,3]]
=> [1,2] => 1 = 0 + 1
[[2,3]]
=> [1,2] => 1 = 0 + 1
[[3,3]]
=> [1,2] => 1 = 0 + 1
[[1],[3]]
=> [2,1] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,4]]
=> [1,2] => 1 = 0 + 1
[[2,4]]
=> [1,2] => 1 = 0 + 1
[[3,4]]
=> [1,2] => 1 = 0 + 1
[[4,4]]
=> [1,2] => 1 = 0 + 1
[[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 = 0 + 1
[[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => 2 = 1 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => 2 = 1 + 1
[[1,5]]
=> [1,2] => 1 = 0 + 1
[[2,5]]
=> [1,2] => 1 = 0 + 1
[[3,5]]
=> [1,2] => 1 = 0 + 1
[[4,5]]
=> [1,2] => 1 = 0 + 1
[[5,5]]
=> [1,2] => 1 = 0 + 1
[[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 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
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1 = 0 + 1
[[2,2]]
=> [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => 2 = 1 + 1
[[1,3]]
=> [1,2] => 1 = 0 + 1
[[2,3]]
=> [1,2] => 1 = 0 + 1
[[3,3]]
=> [1,2] => 1 = 0 + 1
[[1],[3]]
=> [2,1] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,4]]
=> [1,2] => 1 = 0 + 1
[[2,4]]
=> [1,2] => 1 = 0 + 1
[[3,4]]
=> [1,2] => 1 = 0 + 1
[[4,4]]
=> [1,2] => 1 = 0 + 1
[[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 = 0 + 1
[[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => 2 = 1 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => 2 = 1 + 1
[[1,5]]
=> [1,2] => 1 = 0 + 1
[[2,5]]
=> [1,2] => 1 = 0 + 1
[[3,5]]
=> [1,2] => 1 = 0 + 1
[[4,5]]
=> [1,2] => 1 = 0 + 1
[[5,5]]
=> [1,2] => 1 = 0 + 1
[[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 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: St000542
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1 = 0 + 1
[[2,2]]
=> [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => 2 = 1 + 1
[[1,3]]
=> [1,2] => 1 = 0 + 1
[[2,3]]
=> [1,2] => 1 = 0 + 1
[[3,3]]
=> [1,2] => 1 = 0 + 1
[[1],[3]]
=> [2,1] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,4]]
=> [1,2] => 1 = 0 + 1
[[2,4]]
=> [1,2] => 1 = 0 + 1
[[3,4]]
=> [1,2] => 1 = 0 + 1
[[4,4]]
=> [1,2] => 1 = 0 + 1
[[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 = 0 + 1
[[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => 2 = 1 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => 2 = 1 + 1
[[1,5]]
=> [1,2] => 1 = 0 + 1
[[2,5]]
=> [1,2] => 1 = 0 + 1
[[3,5]]
=> [1,2] => 1 = 0 + 1
[[4,5]]
=> [1,2] => 1 = 0 + 1
[[5,5]]
=> [1,2] => 1 = 0 + 1
[[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 left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St001390
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Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001390: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1 = 0 + 1
[[2,2]]
=> [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => 2 = 1 + 1
[[1,3]]
=> [1,2] => 1 = 0 + 1
[[2,3]]
=> [1,2] => 1 = 0 + 1
[[3,3]]
=> [1,2] => 1 = 0 + 1
[[1],[3]]
=> [2,1] => 2 = 1 + 1
[[2],[3]]
=> [2,1] => 2 = 1 + 1
[[1,1,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,2]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[2]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,4]]
=> [1,2] => 1 = 0 + 1
[[2,4]]
=> [1,2] => 1 = 0 + 1
[[3,4]]
=> [1,2] => 1 = 0 + 1
[[4,4]]
=> [1,2] => 1 = 0 + 1
[[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 = 0 + 1
[[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[2,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[3,3,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,1],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[2,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[[1,1,1,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[2,2,2,2]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => 2 = 1 + 1
[[1,2,2],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => 2 = 1 + 1
[[1,5]]
=> [1,2] => 1 = 0 + 1
[[2,5]]
=> [1,2] => 1 = 0 + 1
[[3,5]]
=> [1,2] => 1 = 0 + 1
[[4,5]]
=> [1,2] => 1 = 0 + 1
[[5,5]]
=> [1,2] => 1 = 0 + 1
[[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 bumps occurring when Schensted-inserting the letter 1 of a permutation.
For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
Matching statistic: St000141
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000141: 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] => [1,3,2] => 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] => [1,3,2] => 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 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] => [1,3,2] => 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] => [1,2,4,3] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,2,4] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,3,4,2] => 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 maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000155
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000155: 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] => [3,1,2] => 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] => [3,1,2] => 1
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => 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] => [3,1,2] => 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 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] => [4,1,2,3] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => 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 exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2]
=> [[1,2]]
=> 0
[[2,2]]
=> [2]
=> [[1,2]]
=> 0
[[1],[2]]
=> [1,1]
=> [[1],[2]]
=> 1
[[1,3]]
=> [2]
=> [[1,2]]
=> 0
[[2,3]]
=> [2]
=> [[1,2]]
=> 0
[[3,3]]
=> [2]
=> [[1,2]]
=> 0
[[1],[3]]
=> [1,1]
=> [[1],[2]]
=> 1
[[2],[3]]
=> [1,1]
=> [[1],[2]]
=> 1
[[1,1,2]]
=> [3]
=> [[1,2,3]]
=> 0
[[1,2,2]]
=> [3]
=> [[1,2,3]]
=> 0
[[2,2,2]]
=> [3]
=> [[1,2,3]]
=> 0
[[1,1],[2]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[1,2],[2]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[1,4]]
=> [2]
=> [[1,2]]
=> 0
[[2,4]]
=> [2]
=> [[1,2]]
=> 0
[[3,4]]
=> [2]
=> [[1,2]]
=> 0
[[4,4]]
=> [2]
=> [[1,2]]
=> 0
[[1],[4]]
=> [1,1]
=> [[1],[2]]
=> 1
[[2],[4]]
=> [1,1]
=> [[1],[2]]
=> 1
[[3],[4]]
=> [1,1]
=> [[1],[2]]
=> 1
[[1,1,3]]
=> [3]
=> [[1,2,3]]
=> 0
[[1,2,3]]
=> [3]
=> [[1,2,3]]
=> 0
[[1,3,3]]
=> [3]
=> [[1,2,3]]
=> 0
[[2,2,3]]
=> [3]
=> [[1,2,3]]
=> 0
[[2,3,3]]
=> [3]
=> [[1,2,3]]
=> 0
[[3,3,3]]
=> [3]
=> [[1,2,3]]
=> 0
[[1,1],[3]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[1,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[1,3],[2]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[1,3],[3]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[2,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[2,3],[3]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[[1,1,1,2]]
=> [4]
=> [[1,2,3,4]]
=> 0
[[1,1,2,2]]
=> [4]
=> [[1,2,3,4]]
=> 0
[[1,2,2,2]]
=> [4]
=> [[1,2,3,4]]
=> 0
[[2,2,2,2]]
=> [4]
=> [[1,2,3,4]]
=> 0
[[1,1,1],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> 1
[[1,1,2],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> 1
[[1,2,2],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[1,5]]
=> [2]
=> [[1,2]]
=> 0
[[2,5]]
=> [2]
=> [[1,2]]
=> 0
[[3,5]]
=> [2]
=> [[1,2]]
=> 0
[[4,5]]
=> [2]
=> [[1,2]]
=> 0
[[5,5]]
=> [2]
=> [[1,2]]
=> 0
[[1],[5]]
=> [1,1]
=> [[1],[2]]
=> 1
[[2],[5]]
=> [1,1]
=> [[1],[2]]
=> 1
[[3],[5]]
=> [1,1]
=> [[1],[2]]
=> 1
[[4],[5]]
=> [1,1]
=> [[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 $i$.
The following 315 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000288The number of ones in a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001427The number of descents of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000378The diagonal inversion number of an integer partition. St000527The width of the poset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001389The number of partitions of the same length below the given integer partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000211The rank of the set partition. St000216The absolute length of a permutation. St000234The number of global ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000730The maximal arc length of a set partition. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000956The maximal displacement of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001391The disjunction number of a graph. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. 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. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000702The number of weak deficiencies of a permutation. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000925The number of topologically connected components of a set partition. St000926The clique-coclique number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001058The breadth of the ordered tree. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000094The depth of an ordered tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000806The semiperimeter of the associated bargraph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000006The dinv of a Dyck path. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001896The number of right descents of a signed permutations. 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. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001864The number of excedances of a signed permutation. St001946The number of descents in a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000942The number of critical left to right maxima of the parking functions. 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. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001432The order dimension of the 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. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000928The sum of the coefficients of the character polynomial of an integer 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. 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. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. 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. 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.
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