searching the database
Your data matches 898 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000242
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => 0 = 1 - 1
[[1],[3]]
=> [2,1] => 0 = 1 - 1
[[2],[3]]
=> [2,1] => 0 = 1 - 1
[[1],[4]]
=> [2,1] => 0 = 1 - 1
[[2],[4]]
=> [2,1] => 0 = 1 - 1
[[3],[4]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => 1 = 2 - 1
[[1],[5]]
=> [2,1] => 0 = 1 - 1
[[2],[5]]
=> [2,1] => 0 = 1 - 1
[[3],[5]]
=> [2,1] => 0 = 1 - 1
[[4],[5]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => 1 = 2 - 1
[[1],[6]]
=> [2,1] => 0 = 1 - 1
[[2],[6]]
=> [2,1] => 0 = 1 - 1
[[3],[6]]
=> [2,1] => 0 = 1 - 1
[[4],[6]]
=> [2,1] => 0 = 1 - 1
[[5],[6]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[7]]
=> [2,1] => 0 = 1 - 1
[[2],[7]]
=> [2,1] => 0 = 1 - 1
[[3],[7]]
=> [2,1] => 0 = 1 - 1
[[4],[7]]
=> [2,1] => 0 = 1 - 1
[[5],[7]]
=> [2,1] => 0 = 1 - 1
[[6],[7]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[2],[3],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[2],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[3],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[2],[3],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[8]]
=> [2,1] => 0 = 1 - 1
[[2],[8]]
=> [2,1] => 0 = 1 - 1
[[3],[8]]
=> [2,1] => 0 = 1 - 1
[[4],[8]]
=> [2,1] => 0 = 1 - 1
Description
The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Matching statistic: St000366
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => 0 = 1 - 1
[[1],[3]]
=> [2,1] => 0 = 1 - 1
[[2],[3]]
=> [2,1] => 0 = 1 - 1
[[1],[4]]
=> [2,1] => 0 = 1 - 1
[[2],[4]]
=> [2,1] => 0 = 1 - 1
[[3],[4]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => 1 = 2 - 1
[[1],[5]]
=> [2,1] => 0 = 1 - 1
[[2],[5]]
=> [2,1] => 0 = 1 - 1
[[3],[5]]
=> [2,1] => 0 = 1 - 1
[[4],[5]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => 1 = 2 - 1
[[1],[6]]
=> [2,1] => 0 = 1 - 1
[[2],[6]]
=> [2,1] => 0 = 1 - 1
[[3],[6]]
=> [2,1] => 0 = 1 - 1
[[4],[6]]
=> [2,1] => 0 = 1 - 1
[[5],[6]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[7]]
=> [2,1] => 0 = 1 - 1
[[2],[7]]
=> [2,1] => 0 = 1 - 1
[[3],[7]]
=> [2,1] => 0 = 1 - 1
[[4],[7]]
=> [2,1] => 0 = 1 - 1
[[5],[7]]
=> [2,1] => 0 = 1 - 1
[[6],[7]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[2],[3],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[2],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[3],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[2],[3],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[8]]
=> [2,1] => 0 = 1 - 1
[[2],[8]]
=> [2,1] => 0 = 1 - 1
[[3],[8]]
=> [2,1] => 0 = 1 - 1
[[4],[8]]
=> [2,1] => 0 = 1 - 1
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000371
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => 0 = 1 - 1
[[1],[3]]
=> [2,1] => 0 = 1 - 1
[[2],[3]]
=> [2,1] => 0 = 1 - 1
[[1],[4]]
=> [2,1] => 0 = 1 - 1
[[2],[4]]
=> [2,1] => 0 = 1 - 1
[[3],[4]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => 1 = 2 - 1
[[1],[5]]
=> [2,1] => 0 = 1 - 1
[[2],[5]]
=> [2,1] => 0 = 1 - 1
[[3],[5]]
=> [2,1] => 0 = 1 - 1
[[4],[5]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => 1 = 2 - 1
[[1],[6]]
=> [2,1] => 0 = 1 - 1
[[2],[6]]
=> [2,1] => 0 = 1 - 1
[[3],[6]]
=> [2,1] => 0 = 1 - 1
[[4],[6]]
=> [2,1] => 0 = 1 - 1
[[5],[6]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[7]]
=> [2,1] => 0 = 1 - 1
[[2],[7]]
=> [2,1] => 0 = 1 - 1
[[3],[7]]
=> [2,1] => 0 = 1 - 1
[[4],[7]]
=> [2,1] => 0 = 1 - 1
[[5],[7]]
=> [2,1] => 0 = 1 - 1
[[6],[7]]
=> [2,1] => 0 = 1 - 1
[[1],[2],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => 1 = 2 - 1
[[1],[2],[3],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[2],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[3],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[2],[3],[4],[5]]
=> [4,3,2,1] => 2 = 3 - 1
[[1],[8]]
=> [2,1] => 0 = 1 - 1
[[2],[8]]
=> [2,1] => 0 = 1 - 1
[[3],[8]]
=> [2,1] => 0 = 1 - 1
[[4],[8]]
=> [2,1] => 0 = 1 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000003
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [[1,2]]
=> [2]
=> 1
[[1],[3]]
=> [[1,3]]
=> [2]
=> 1
[[2],[3]]
=> [[2,3]]
=> [2]
=> 1
[[1],[4]]
=> [[1,4]]
=> [2]
=> 1
[[2],[4]]
=> [[2,4]]
=> [2]
=> 1
[[3],[4]]
=> [[3,4]]
=> [2]
=> 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> 2
[[1],[5]]
=> [[1,5]]
=> [2]
=> 1
[[2],[5]]
=> [[2,5]]
=> [2]
=> 1
[[3],[5]]
=> [[3,5]]
=> [2]
=> 1
[[4],[5]]
=> [[4,5]]
=> [2]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> 2
[[1],[6]]
=> [[1,6]]
=> [2]
=> 1
[[2],[6]]
=> [[2,6]]
=> [2]
=> 1
[[3],[6]]
=> [[3,6]]
=> [2]
=> 1
[[4],[6]]
=> [[4,6]]
=> [2]
=> 1
[[5],[6]]
=> [[5,6]]
=> [2]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> 3
[[1],[7]]
=> [[1,7]]
=> [2]
=> 1
[[2],[7]]
=> [[2,7]]
=> [2]
=> 1
[[3],[7]]
=> [[3,7]]
=> [2]
=> 1
[[4],[7]]
=> [[4,7]]
=> [2]
=> 1
[[5],[7]]
=> [[5,7]]
=> [2]
=> 1
[[6],[7]]
=> [[6,7]]
=> [2]
=> 1
[[1],[2],[6]]
=> [[1,2],[6]]
=> [2,1]
=> 2
[[1],[3],[6]]
=> [[1,3],[6]]
=> [2,1]
=> 2
[[1],[4],[6]]
=> [[1,4],[6]]
=> [2,1]
=> 2
[[1],[5],[6]]
=> [[1,5],[6]]
=> [2,1]
=> 2
[[2],[3],[6]]
=> [[2,3],[6]]
=> [2,1]
=> 2
[[2],[4],[6]]
=> [[2,4],[6]]
=> [2,1]
=> 2
[[2],[5],[6]]
=> [[2,5],[6]]
=> [2,1]
=> 2
[[3],[4],[6]]
=> [[3,4],[6]]
=> [2,1]
=> 2
[[3],[5],[6]]
=> [[3,5],[6]]
=> [2,1]
=> 2
[[4],[5],[6]]
=> [[4,5],[6]]
=> [2,1]
=> 2
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> 3
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> 3
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> 3
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> 3
[[1],[8]]
=> [[1,8]]
=> [2]
=> 1
[[2],[8]]
=> [[2,8]]
=> [2]
=> 1
[[3],[8]]
=> [[3,8]]
=> [2]
=> 1
[[4],[8]]
=> [[4,8]]
=> [2]
=> 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000007
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [[1,2]]
=> [1,2] => 1
[[1],[3]]
=> [[1,3]]
=> [1,2] => 1
[[2],[3]]
=> [[2,3]]
=> [1,2] => 1
[[1],[4]]
=> [[1,4]]
=> [1,2] => 1
[[2],[4]]
=> [[2,4]]
=> [1,2] => 1
[[3],[4]]
=> [[3,4]]
=> [1,2] => 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[1],[5]]
=> [[1,5]]
=> [1,2] => 1
[[2],[5]]
=> [[2,5]]
=> [1,2] => 1
[[3],[5]]
=> [[3,5]]
=> [1,2] => 1
[[4],[5]]
=> [[4,5]]
=> [1,2] => 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [3,1,2] => 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [3,1,2] => 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [3,1,2] => 2
[[1],[6]]
=> [[1,6]]
=> [1,2] => 1
[[2],[6]]
=> [[2,6]]
=> [1,2] => 1
[[3],[6]]
=> [[3,6]]
=> [1,2] => 1
[[4],[6]]
=> [[4,6]]
=> [1,2] => 1
[[5],[6]]
=> [[5,6]]
=> [1,2] => 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [3,1,2] => 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [3,1,2] => 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [3,1,2] => 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [3,1,2] => 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [3,1,2] => 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [3,1,2] => 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3
[[1],[7]]
=> [[1,7]]
=> [1,2] => 1
[[2],[7]]
=> [[2,7]]
=> [1,2] => 1
[[3],[7]]
=> [[3,7]]
=> [1,2] => 1
[[4],[7]]
=> [[4,7]]
=> [1,2] => 1
[[5],[7]]
=> [[5,7]]
=> [1,2] => 1
[[6],[7]]
=> [[6,7]]
=> [1,2] => 1
[[1],[2],[6]]
=> [[1,2],[6]]
=> [3,1,2] => 2
[[1],[3],[6]]
=> [[1,3],[6]]
=> [3,1,2] => 2
[[1],[4],[6]]
=> [[1,4],[6]]
=> [3,1,2] => 2
[[1],[5],[6]]
=> [[1,5],[6]]
=> [3,1,2] => 2
[[2],[3],[6]]
=> [[2,3],[6]]
=> [3,1,2] => 2
[[2],[4],[6]]
=> [[2,4],[6]]
=> [3,1,2] => 2
[[2],[5],[6]]
=> [[2,5],[6]]
=> [3,1,2] => 2
[[3],[4],[6]]
=> [[3,4],[6]]
=> [3,1,2] => 2
[[3],[5],[6]]
=> [[3,5],[6]]
=> [3,1,2] => 2
[[4],[5],[6]]
=> [[4,5],[6]]
=> [3,1,2] => 2
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [4,3,1,2] => 3
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [4,3,1,2] => 3
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [4,3,1,2] => 3
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [4,3,1,2] => 3
[[1],[8]]
=> [[1,8]]
=> [1,2] => 1
[[2],[8]]
=> [[2,8]]
=> [1,2] => 1
[[3],[8]]
=> [[3,8]]
=> [1,2] => 1
[[4],[8]]
=> [[4,8]]
=> [1,2] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000010
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [[1,2]]
=> [2]
=> 1
[[1],[3]]
=> [[1,3]]
=> [2]
=> 1
[[2],[3]]
=> [[2,3]]
=> [2]
=> 1
[[1],[4]]
=> [[1,4]]
=> [2]
=> 1
[[2],[4]]
=> [[2,4]]
=> [2]
=> 1
[[3],[4]]
=> [[3,4]]
=> [2]
=> 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> 2
[[1],[5]]
=> [[1,5]]
=> [2]
=> 1
[[2],[5]]
=> [[2,5]]
=> [2]
=> 1
[[3],[5]]
=> [[3,5]]
=> [2]
=> 1
[[4],[5]]
=> [[4,5]]
=> [2]
=> 1
[[1],[2],[4]]
=> [[1,2],[4]]
=> [2,1]
=> 2
[[1],[3],[4]]
=> [[1,3],[4]]
=> [2,1]
=> 2
[[2],[3],[4]]
=> [[2,3],[4]]
=> [2,1]
=> 2
[[1],[6]]
=> [[1,6]]
=> [2]
=> 1
[[2],[6]]
=> [[2,6]]
=> [2]
=> 1
[[3],[6]]
=> [[3,6]]
=> [2]
=> 1
[[4],[6]]
=> [[4,6]]
=> [2]
=> 1
[[5],[6]]
=> [[5,6]]
=> [2]
=> 1
[[1],[2],[5]]
=> [[1,2],[5]]
=> [2,1]
=> 2
[[1],[3],[5]]
=> [[1,3],[5]]
=> [2,1]
=> 2
[[1],[4],[5]]
=> [[1,4],[5]]
=> [2,1]
=> 2
[[2],[3],[5]]
=> [[2,3],[5]]
=> [2,1]
=> 2
[[2],[4],[5]]
=> [[2,4],[5]]
=> [2,1]
=> 2
[[3],[4],[5]]
=> [[3,4],[5]]
=> [2,1]
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> 3
[[1],[7]]
=> [[1,7]]
=> [2]
=> 1
[[2],[7]]
=> [[2,7]]
=> [2]
=> 1
[[3],[7]]
=> [[3,7]]
=> [2]
=> 1
[[4],[7]]
=> [[4,7]]
=> [2]
=> 1
[[5],[7]]
=> [[5,7]]
=> [2]
=> 1
[[6],[7]]
=> [[6,7]]
=> [2]
=> 1
[[1],[2],[6]]
=> [[1,2],[6]]
=> [2,1]
=> 2
[[1],[3],[6]]
=> [[1,3],[6]]
=> [2,1]
=> 2
[[1],[4],[6]]
=> [[1,4],[6]]
=> [2,1]
=> 2
[[1],[5],[6]]
=> [[1,5],[6]]
=> [2,1]
=> 2
[[2],[3],[6]]
=> [[2,3],[6]]
=> [2,1]
=> 2
[[2],[4],[6]]
=> [[2,4],[6]]
=> [2,1]
=> 2
[[2],[5],[6]]
=> [[2,5],[6]]
=> [2,1]
=> 2
[[3],[4],[6]]
=> [[3,4],[6]]
=> [2,1]
=> 2
[[3],[5],[6]]
=> [[3,5],[6]]
=> [2,1]
=> 2
[[4],[5],[6]]
=> [[4,5],[6]]
=> [2,1]
=> 2
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> 3
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> 3
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> 3
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> 3
[[1],[8]]
=> [[1,8]]
=> [2]
=> 1
[[2],[8]]
=> [[2,8]]
=> [2]
=> 1
[[3],[8]]
=> [[3,8]]
=> [2]
=> 1
[[4],[8]]
=> [[4,8]]
=> [2]
=> 1
Description
The length of the partition.
Matching statistic: St000015
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[6]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[6]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[6]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4],[6]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[5],[6]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1],[7]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[7]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[7]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4],[7]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[5],[7]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[6],[7]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[3],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[3],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[4],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1],[8]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[8]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[8]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4],[8]]
=> [1,1]
=> [1,1,0,0]
=> 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000054
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => [1,2] => 1
[[1],[3]]
=> [2,1] => [1,2] => 1
[[2],[3]]
=> [2,1] => [1,2] => 1
[[1],[4]]
=> [2,1] => [1,2] => 1
[[2],[4]]
=> [2,1] => [1,2] => 1
[[3],[4]]
=> [2,1] => [1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [2,1,3] => 2
[[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
[[1],[2],[4]]
=> [3,2,1] => [2,1,3] => 2
[[1],[3],[4]]
=> [3,2,1] => [2,1,3] => 2
[[2],[3],[4]]
=> [3,2,1] => [2,1,3] => 2
[[1],[6]]
=> [2,1] => [1,2] => 1
[[2],[6]]
=> [2,1] => [1,2] => 1
[[3],[6]]
=> [2,1] => [1,2] => 1
[[4],[6]]
=> [2,1] => [1,2] => 1
[[5],[6]]
=> [2,1] => [1,2] => 1
[[1],[2],[5]]
=> [3,2,1] => [2,1,3] => 2
[[1],[3],[5]]
=> [3,2,1] => [2,1,3] => 2
[[1],[4],[5]]
=> [3,2,1] => [2,1,3] => 2
[[2],[3],[5]]
=> [3,2,1] => [2,1,3] => 2
[[2],[4],[5]]
=> [3,2,1] => [2,1,3] => 2
[[3],[4],[5]]
=> [3,2,1] => [2,1,3] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,1,4] => 3
[[1],[7]]
=> [2,1] => [1,2] => 1
[[2],[7]]
=> [2,1] => [1,2] => 1
[[3],[7]]
=> [2,1] => [1,2] => 1
[[4],[7]]
=> [2,1] => [1,2] => 1
[[5],[7]]
=> [2,1] => [1,2] => 1
[[6],[7]]
=> [2,1] => [1,2] => 1
[[1],[2],[6]]
=> [3,2,1] => [2,1,3] => 2
[[1],[3],[6]]
=> [3,2,1] => [2,1,3] => 2
[[1],[4],[6]]
=> [3,2,1] => [2,1,3] => 2
[[1],[5],[6]]
=> [3,2,1] => [2,1,3] => 2
[[2],[3],[6]]
=> [3,2,1] => [2,1,3] => 2
[[2],[4],[6]]
=> [3,2,1] => [2,1,3] => 2
[[2],[5],[6]]
=> [3,2,1] => [2,1,3] => 2
[[3],[4],[6]]
=> [3,2,1] => [2,1,3] => 2
[[3],[5],[6]]
=> [3,2,1] => [2,1,3] => 2
[[4],[5],[6]]
=> [3,2,1] => [2,1,3] => 2
[[1],[2],[3],[5]]
=> [4,3,2,1] => [3,2,1,4] => 3
[[1],[2],[4],[5]]
=> [4,3,2,1] => [3,2,1,4] => 3
[[1],[3],[4],[5]]
=> [4,3,2,1] => [3,2,1,4] => 3
[[2],[3],[4],[5]]
=> [4,3,2,1] => [3,2,1,4] => 3
[[1],[8]]
=> [2,1] => [1,2] => 1
[[2],[8]]
=> [2,1] => [1,2] => 1
[[3],[8]]
=> [2,1] => [1,2] => 1
[[4],[8]]
=> [2,1] => [1,2] => 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000062
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => [2,1] => 1
[[1],[3]]
=> [2,1] => [2,1] => 1
[[2],[3]]
=> [2,1] => [2,1] => 1
[[1],[4]]
=> [2,1] => [2,1] => 1
[[2],[4]]
=> [2,1] => [2,1] => 1
[[3],[4]]
=> [2,1] => [2,1] => 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 2
[[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
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => 2
[[1],[6]]
=> [2,1] => [2,1] => 1
[[2],[6]]
=> [2,1] => [2,1] => 1
[[3],[6]]
=> [2,1] => [2,1] => 1
[[4],[6]]
=> [2,1] => [2,1] => 1
[[5],[6]]
=> [2,1] => [2,1] => 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[7]]
=> [2,1] => [2,1] => 1
[[2],[7]]
=> [2,1] => [2,1] => 1
[[3],[7]]
=> [2,1] => [2,1] => 1
[[4],[7]]
=> [2,1] => [2,1] => 1
[[5],[7]]
=> [2,1] => [2,1] => 1
[[6],[7]]
=> [2,1] => [2,1] => 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => 2
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => 2
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => 2
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[2],[3],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[2],[4],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[3],[4],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[2],[3],[4],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[8]]
=> [2,1] => [2,1] => 1
[[2],[8]]
=> [2,1] => [2,1] => 1
[[3],[8]]
=> [2,1] => [2,1] => 1
[[4],[8]]
=> [2,1] => [2,1] => 1
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000213
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [2,1] => [2,1] => 1
[[1],[3]]
=> [2,1] => [2,1] => 1
[[2],[3]]
=> [2,1] => [2,1] => 1
[[1],[4]]
=> [2,1] => [2,1] => 1
[[2],[4]]
=> [2,1] => [2,1] => 1
[[3],[4]]
=> [2,1] => [2,1] => 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 2
[[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
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => 2
[[1],[6]]
=> [2,1] => [2,1] => 1
[[2],[6]]
=> [2,1] => [2,1] => 1
[[3],[6]]
=> [2,1] => [2,1] => 1
[[4],[6]]
=> [2,1] => [2,1] => 1
[[5],[6]]
=> [2,1] => [2,1] => 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[7]]
=> [2,1] => [2,1] => 1
[[2],[7]]
=> [2,1] => [2,1] => 1
[[3],[7]]
=> [2,1] => [2,1] => 1
[[4],[7]]
=> [2,1] => [2,1] => 1
[[5],[7]]
=> [2,1] => [2,1] => 1
[[6],[7]]
=> [2,1] => [2,1] => 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => 2
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => 2
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => 2
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => 2
[[1],[2],[3],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[2],[4],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[3],[4],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[2],[3],[4],[5]]
=> [4,3,2,1] => [2,3,4,1] => 3
[[1],[8]]
=> [2,1] => [2,1] => 1
[[2],[8]]
=> [2,1] => [2,1] => 1
[[3],[8]]
=> [2,1] => [2,1] => 1
[[4],[8]]
=> [2,1] => [2,1] => 1
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
The following 888 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000239The number of small weak excedances. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000531The leading coefficient of the rook polynomial of an integer partition. St000542The number of left-to-right-minima of a permutation. St000617The number of global maxima of a Dyck path. St000638The number of up-down runs of a permutation. St000740The last entry of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000820The number of compositions obtained by rotating the composition. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000991The number of right-to-left minima of a permutation. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 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. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001497The position of the largest weak excedence of a permutation. St001652The length of a longest interval of consecutive numbers. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000002The number of occurrences of the pattern 123 in a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000377The dinv defect of an integer partition. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000546The number of global descents of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000731The number of double exceedences of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000931The number of occurrences of the pattern UUU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001096The size of the overlap set of a permutation. St001160The number of proper blocks (or intervals) of a permutations. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001298The number of repeated entries in the Lehmer code of a permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001377The major index minus the number of inversions of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001911A descent variant minus the number of inversions. St000001The number of reduced words for a permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000047The number of standard immaculate tableaux of a given shape. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000075The orbit size of a standard tableau under promotion. St000078The number of alternating sign matrices whose left key is the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000184The size of the centralizer of any permutation of given cycle type. St000240The number of indices that are not small excedances. St000255The number of reduced Kogan faces with the permutation as type. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000277The number of ribbon shaped standard tableaux. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000528The height of a poset. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000553The number of blocks of a graph. St000626The minimal period of a binary word. 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. St000686The finitistic dominant dimension of a Dyck path. St000703The number of deficiencies of a permutation. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000759The smallest missing part in an integer partition. St000765The number of weak records in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000822The Hadwiger number of the graph. St000839The largest opener of a set partition. St000874The position of the last double rise in a Dyck path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000907The number of maximal antichains of minimal length in a poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001050The number of terminal closers of a set partition. St001058The breadth of the ordered tree. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001268The size of the largest ordinal summand in the poset. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in 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. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001343The dimension of the reduced incidence algebra of a poset. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001399The distinguishing number of a poset. St001400The total number of Littlewood-Richardson tableaux of given shape. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001595The number of standard Young tableaux of the skew partition. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001779The order of promotion on the set of linear extensions of a poset. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000029The depth of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000050The depth or height of a binary tree. St000051The size of the left subtree of a binary tree. St000057The Shynar inversion number of a standard tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000080The rank of the poset. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000133The "bounce" of a permutation. St000148The number of odd parts of a partition. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000204The number of internal nodes of a binary tree. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000224The sorting index of a permutation. St000225Difference between largest and smallest parts in a partition. St000228The size of a partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000246The number of non-inversions of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000304The load of a permutation. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000313The number of degree 2 vertices of a graph. 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. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000357The number of occurrences of the pattern 12-3. St000362The size of a minimal vertex cover of a graph. St000367The number of simsun double descents of a permutation. St000369The dinv deficit of a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000376The bounce deficit of a Dyck path. St000384The maximal part of the shifted composition of an integer partition. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000496The rcs statistic of a set partition. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000766The number of inversions of an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000867The sum of the hook lengths in the first row of an integer partition. St000921The number of internal inversions of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001127The sum of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001427The number of descents of a signed permutation. St001459The number of zero columns in the nullspace of a graph. St001484The number of singletons of an integer partition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001521Half the total irregularity of a graph. St001554The number of distinct nonempty subtrees of a binary tree. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001644The dimension of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001671Haglund's hag of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001726The number of visible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001759The Rajchgot index of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001781The interlacing number of a set partition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001812The biclique partition number of a graph. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001910The height of the middle non-run of a Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001721The degree of a binary word. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000702The number of weak deficiencies of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000889The number of alternating sign matrices with the same antidiagonal sums. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001405The number of bonds in a permutation. St001480The number of simple summands of the module J^2/J^3. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St000064The number of one-box pattern of a permutation. St000144The pyramid weight of the Dyck path. St000236The number of cyclical small weak excedances. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000989The number of final rises of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001516The number of cyclic bonds of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001684The reduced word complexity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001822The number of alignments of a signed permutation. St001866The nesting alignments of a signed permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000942The number of critical left to right maxima of the parking functions. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001510The number of self-evacuating linear extensions of a finite poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001863The number of weak excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000956The maximal displacement of a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001712The number of natural descents of a standard Young tableau. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. 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. St001946The number of descents in a parking function. St001948The number of augmented double ascents of a permutation. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000060The greater neighbor of the maximum. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001052The length of the exterior of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St000625The sum of the minimal distances to a greater element. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001074The number of inversions of the cyclic embedding of a permutation. St000520The number of patterns in a permutation. St000216The absolute length of a permutation. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000385The number of vertices with out-degree 1 in a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000460The hook length of the last cell along the main diagonal of an integer partition. St000495The number of inversions of distance at most 2 of a permutation. St000503The maximal difference between two elements in a common block. St000530The number of permutations with the same descent word as the given permutation. St000539The number of odd inversions of a permutation. St000627The exponent of a binary word. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000730The maximal arc length of a set partition. St000744The length of the path to the largest entry in a standard Young tableau. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000833The comajor index of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000886The number of permutations with the same antidiagonal sums. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001267The length of the Lyndon factorization of the binary word. St001360The number of covering relations in Young's lattice below a partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St001933The largest multiplicity of a part in an integer partition. St000061The number of nodes on the left branch of a binary tree. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000294The number of distinct factors of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000438The position of the last up step in a Dyck path. St000461The rix statistic of a permutation. St000471The sum of the ascent tops of a permutation. St000477The weight of a partition according to Alladi. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000518The number of distinct subsequences in a binary word. St000538The number of even inversions of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000656The number of cuts of a poset. St000668The least common multiple of the parts of the partition. St000673The number of non-fixed points of a permutation. St000675The number of centered multitunnels of a Dyck path. St000680The Grundy value for Hackendot on posets. St000708The product of the parts of an integer partition. St000710The number of big deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000770The major index of an integer partition when read from bottom to top. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000961The shifted major index of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001346The number of parking functions that give the same permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001388The number of non-attacking neighbors of a permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001959The product of the heights of the peaks of a Dyck path. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000806The semiperimeter of the associated bargraph. St000083The number of left oriented leafs of a binary tree except the first one. St000087The number of induced subgraphs. St000117The number of centered tunnels of a Dyck path. St000134The size of the orbit of an alternating sign matrix under gyration. St000189The number of elements in the poset. St000197The number of entries equal to positive one in the alternating sign matrix. St000203The number of external nodes of a binary tree. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000338The number of pixed points of a permutation. St000391The sum of the positions of the ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000469The distinguishing number of a graph. St000472The sum of the ascent bottoms of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. St000567The sum of the products of all pairs of parts. St000636The hull number of a graph. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000667The greatest common divisor of the parts of the partition. St000693The modular (standard) major index of a standard tableau. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000728The dimension of a set partition. St000729The minimal arc length of a set partition. St000849The number of 1/3-balanced pairs in a poset. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000925The number of topologically connected components of a set partition. St000926The clique-coclique number of a graph. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001118The acyclic chromatic index of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001128The exponens consonantiae of a partition. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001342The number of vertices in the center of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001430The number of positive entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001439The number of even weak deficiencies and of odd weak exceedences. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. 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. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001675The number of parts equal to the part in the reversed composition. 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. St001746The coalition number of a graph. St001807The lower middle entry of a permutation. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000014The number of parking functions supported by a Dyck path. St000082The number of elements smaller than a binary tree in Tamari order. St000100The number of linear extensions of a poset. St000145The Dyson rank of a partition. St000180The number of chains of a poset. St000210Minimum over maximum difference of elements in cycles. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000301The number of facets of the stable set polytope of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000674The number of hills of a Dyck path. St000691The number of changes of a binary word. St000696The number of cycles in the breakpoint graph of a permutation. St000711The number of big exceedences of a permutation. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000823The number of unsplittable factors of the set partition. St000890The number of nonzero entries in an alternating sign matrix. St000894The trace of an alternating sign matrix. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001119The length of a shortest maximal path in a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001357The maximal degree of a regular spanning subgraph of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001552The number of inversions between excedances and fixed points of a permutation. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001664The number of non-isomorphic subposets of a poset. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001808The box weight or horizontal decoration of a Dyck path. St001925The minimal number of zeros in a row of an alternating sign matrix. St001949The rigidity index of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001958The degree of the polynomial interpolating the values of a permutation. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000327The number of cover relations in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001811The Castelnuovo-Mumford regularity of a permutation. St000045The number of linear extensions of a binary tree. St000227The osculating paths major index of an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001927Sparre Andersen's number of positives of a signed permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001854The size of the left Kazhdan-Lusztig cell, St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001857The number of edges in the reduced word graph of a signed permutation. St000307The number of rowmotion orbits of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001626The number of maximal proper sublattices of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001926Sparre Andersen's position of the maximum of a signed permutation. 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. St001875The number of simple modules with projective dimension at most 1. 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. St000219The number of occurrences of the pattern 231 in a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000937The number of positive values of the symmetric group character corresponding to the partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001432The order dimension of the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000677The standardized bi-alternating inversion number of a permutation. St001280The number of parts of an integer partition that are at least two. St000474Dyson's crank of a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000782The indicator function of whether a given perfect matching is an L & P matching. St001890The maximum magnitude of the Möbius function of a poset. St000102The charge of a semistandard tableau. St000422The energy of a graph, if it is integral. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000456The monochromatic index of a connected graph. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001877Number of indecomposable injective modules with projective dimension 2. St000464The Schultz index of a connected graph. St001407The number of minimal entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!