Your data matches 509 different statistics following compositions of up to 3 maps.
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St000319: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 2
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 3
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 4
[4,1]
=> 3
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 5
[5,1]
=> 4
[4,2]
=> 3
[4,1,1]
=> 3
[3,3]
=> 3
[3,2,1]
=> 2
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
St000320: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 2
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 3
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 4
[4,1]
=> 3
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 5
[5,1]
=> 4
[4,2]
=> 3
[4,1,1]
=> 3
[3,3]
=> 3
[3,2,1]
=> 2
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 0
[2]
=> [[1,2]]
=> [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 3
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5
Description
The number of circled entries of the shifted recording tableau of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of circled entries in $Q$.
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 5
Description
The number of visible inversions of a permutation. A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3 = 2 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4 = 3 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 4 = 3 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 6 = 5 + 1
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3 = 2 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 4 = 3 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4 = 3 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 6 = 5 + 1
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 3
Description
The number of stack-sorts needed to sort a permutation. A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series. Let $W_t(n,k)$ be the number of permutations of size $n$ with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 4
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 5
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000154
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,4,5] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1
Description
The sum of the descent bottoms of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$ For the descent tops, see [[St000111]].
The following 499 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000293The number of inversions of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000507The number of ascents of a standard tableau. St000868The aid statistic in the sense of Shareshian-Wachs. St000921The number of internal inversions of a binary word. St001090The number of pop-stack-sorts needed to sort a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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 $eA$. St001215Let 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000019The cardinality of the support of a permutation. St000054The first entry of the permutation. St000147The largest part of an integer partition. St000240The number of indices that are not small excedances. St000676The number of odd rises of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000776The maximal multiplicity of an eigenvalue in a graph. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. 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. St000877The depth of the binary word interpreted as a path. St000485The length of the longest cycle of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001727The number of invisible inversions of a permutation. St000006The dinv of a Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001571The Cartan determinant of the integer partition. St000011The number of touch points (or returns) of a Dyck path. St001812The biclique partition number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000306The bounce count of a Dyck path. St001330The hat guessing number of a graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000647The number of big descents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000031The number of cycles in the cycle decomposition of a permutation. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000532The total number of rook placements on a Ferrers board. St000710The number of big deficiencies of a permutation. St000738The first entry in the last row of a standard tableau. St001152The number of pairs with even minimum in a perfect matching. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000026The position of the first return of a Dyck path. St000060The greater neighbor of the maximum. St000083The number of left oriented leafs of a binary tree except the first one. St000110The number of permutations less than or equal to a permutation in left weak order. St000133The "bounce" of a permutation. St000296The length of the symmetric border of a binary word. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000381The largest part of an integer composition. St000383The last part of an integer composition. St000385The number of vertices with out-degree 1 in a binary tree. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St000719The number of alignments in a perfect matching. St000734The last entry in the first row of a standard tableau. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St000885The number of critical steps in the Catalan decomposition of a binary word. St000989The number of final rises 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. 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. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. 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. 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001684The reduced word complexity of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001933The largest multiplicity of a part in an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000015The number of peaks of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000089The absolute variation of a composition. St000155The number of exceedances (also excedences) of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000443The number of long tunnels of a Dyck path. St000536The pathwidth of a graph. St000702The number of weak deficiencies of a permutation. St000809The reduced reflection length of the permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St000061The number of nodes on the left branch of a binary tree. St000093The cardinality of a maximal independent set of vertices of a graph. St000098The chromatic number of a graph. St000144The pyramid weight of the Dyck path. St000172The Grundy number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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: St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001746The coalition number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000454The largest eigenvalue of a graph if it is integral. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001637The number of (upper) dissectors of a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001638The book thickness of a graph. St000736The last entry in the first row of a semistandard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000045The number of linear extensions of a binary tree. St000420The number of Dyck paths that are weakly above a Dyck path. St000487The length of the shortest cycle of a permutation. St000729The minimal arc length of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000990The first ascent of a permutation. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000181The number of connected components of the Hasse diagram for the poset. St001644The dimension of a graph. St000628The balance of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000682The Grundy value of Welter's game on a binary word. 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. St001557The number of inversions of the second entry of a permutation. St001712The number of natural descents of a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001937The size of the center of a parking function. St000822The Hadwiger number of the graph. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000815The number of semistandard Young tableaux of partition weight of given shape. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000010The length of the partition. St000014The number of parking functions supported by a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000288The number of ones in a binary word. St000290The major index of a binary word. St000297The number of leading ones in a binary word. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000475The number of parts equal to 1 in a partition. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000733The row containing the largest entry of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000867The sum of the hook lengths in the first row of an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St001018Sum of projective dimension of the indecomposable injective modules of 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. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. 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. St001527The cyclic permutation representation number of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001658The total number of rook placements on a Ferrers board. St001884The number of borders of a binary word. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000005The bounce statistic of a Dyck path. St000766The number of inversions of an integer composition. St000840The number of closers smaller than the largest opener in a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001863The number of weak excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000740The last entry of a permutation. St001424The number of distinct squares in a binary word. St001488The number of corners of a skew partition. St000691The number of changes of a binary word. St000735The last entry on the main diagonal of a standard tableau. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000993The multiplicity of the largest part of an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001896The number of right descents of a signed permutations. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001946The number of descents in a parking function. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000456The monochromatic index of a connected graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000374The number of exclusive right-to-left minima of a permutation. St001096The size of the overlap set of a permutation. 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 $eA$. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000662The staircase size of the code of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000478Another weight of a partition according to Alladi. St000678The number of up steps after the last double rise of a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000984The number of boxes below precisely one peak. 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. St001128The exponens consonantiae of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001378The product of the cohook lengths of the integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000091The descent variation of a composition. St000137The Grundy value of an integer partition. St000173The segment statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000491The number of inversions of a set partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. 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. St000650The number of 3-rises of a permutation. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001403The number of vertical separators in a permutation. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001568The smallest positive integer that does not appear twice in the partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001843The Z-index of a set partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000314The number of left-to-right-maxima of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000707The product of the factorials of the parts. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000834The number of right outer peaks of a permutation. St000007The number of saliances of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000455The second largest eigenvalue of a graph if it is integral. St001487The number of inner corners of a skew partition. 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. St000264The girth of a graph, which is not a tree. St000365The number of double ascents of a permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001060The distinguishing index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to 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 $eA$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000023The number of inner peaks of a permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000779The tier of a permutation. St001469The holeyness of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001935The number of ascents in a parking function. St000075The orbit size of a standard tableau under promotion. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001811The Castelnuovo-Mumford regularity of a permutation. St000632The jump number of the poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St000717The number of ordinal summands of a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral.