Your data matches 666 different statistics following compositions of up to 3 maps.
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St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 2
[2,2]
=> 2
[2,1,1]
=> 3
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 2
[3,1,1]
=> 3
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 2
[4,2]
=> 2
[4,1,1]
=> 3
[3,3]
=> 2
[3,1,1,1]
=> 4
[2,1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> 6
[2,1,1,1,1,1]
=> 6
[1,1,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1,1]
=> 8
Description
The length of the partition.
Mp00044: Integer partitions conjugateInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 2
[3]
=> [1,1,1]
=> 1
[2,1]
=> [2,1]
=> 2
[1,1,1]
=> [3]
=> 3
[4]
=> [1,1,1,1]
=> 1
[3,1]
=> [2,1,1]
=> 2
[2,2]
=> [2,2]
=> 2
[2,1,1]
=> [3,1]
=> 3
[1,1,1,1]
=> [4]
=> 4
[5]
=> [1,1,1,1,1]
=> 1
[4,1]
=> [2,1,1,1]
=> 2
[3,2]
=> [2,2,1]
=> 2
[3,1,1]
=> [3,1,1]
=> 3
[2,1,1,1]
=> [4,1]
=> 4
[1,1,1,1,1]
=> [5]
=> 5
[6]
=> [1,1,1,1,1,1]
=> 1
[5,1]
=> [2,1,1,1,1]
=> 2
[4,2]
=> [2,2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> 3
[3,3]
=> [2,2,2]
=> 2
[3,1,1,1]
=> [4,1,1]
=> 4
[2,1,1,1,1]
=> [5,1]
=> 5
[1,1,1,1,1,1]
=> [6]
=> 6
[2,1,1,1,1,1]
=> [6,1]
=> 6
[1,1,1,1,1,1,1]
=> [7]
=> 7
[1,1,1,1,1,1,1,1]
=> [8]
=> 8
Description
The largest part of an integer partition.
Mp00095: Integer partitions to binary wordBinary words
St000288: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 1
[2]
=> 100 => 1
[1,1]
=> 110 => 2
[3]
=> 1000 => 1
[2,1]
=> 1010 => 2
[1,1,1]
=> 1110 => 3
[4]
=> 10000 => 1
[3,1]
=> 10010 => 2
[2,2]
=> 1100 => 2
[2,1,1]
=> 10110 => 3
[1,1,1,1]
=> 11110 => 4
[5]
=> 100000 => 1
[4,1]
=> 100010 => 2
[3,2]
=> 10100 => 2
[3,1,1]
=> 100110 => 3
[2,1,1,1]
=> 101110 => 4
[1,1,1,1,1]
=> 111110 => 5
[6]
=> 1000000 => 1
[5,1]
=> 1000010 => 2
[4,2]
=> 100100 => 2
[4,1,1]
=> 1000110 => 3
[3,3]
=> 11000 => 2
[3,1,1,1]
=> 1001110 => 4
[2,1,1,1,1]
=> 1011110 => 5
[1,1,1,1,1,1]
=> 1111110 => 6
[2,1,1,1,1,1]
=> 10111110 => 6
[1,1,1,1,1,1,1]
=> 11111110 => 7
[1,1,1,1,1,1,1,1]
=> 111111110 => 8
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 2
[3]
=> [1,1,1]
=> 1
[2,1]
=> [3]
=> 2
[1,1,1]
=> [2,1]
=> 3
[4]
=> [1,1,1,1]
=> 1
[3,1]
=> [2,1,1]
=> 2
[2,2]
=> [4]
=> 2
[2,1,1]
=> [2,2]
=> 3
[1,1,1,1]
=> [3,1]
=> 4
[5]
=> [1,1,1,1,1]
=> 1
[4,1]
=> [2,1,1,1]
=> 2
[3,2]
=> [5]
=> 2
[3,1,1]
=> [4,1]
=> 3
[2,1,1,1]
=> [3,1,1]
=> 4
[1,1,1,1,1]
=> [3,2]
=> 5
[6]
=> [1,1,1,1,1,1]
=> 1
[5,1]
=> [2,1,1,1,1]
=> 2
[4,2]
=> [2,2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> 3
[3,3]
=> [6]
=> 2
[3,1,1,1]
=> [3,3]
=> 4
[2,1,1,1,1]
=> [4,2]
=> 5
[1,1,1,1,1,1]
=> [3,2,1]
=> 6
[2,1,1,1,1,1]
=> [3,3,1]
=> 6
[1,1,1,1,1,1,1]
=> [4,2,1]
=> 7
[1,1,1,1,1,1,1,1]
=> [4,3,1]
=> 8
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Mp00042: Integer partitions initial tableauStandard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> 1
[2]
=> [[1,2]]
=> 1
[1,1]
=> [[1],[2]]
=> 2
[3]
=> [[1,2,3]]
=> 1
[2,1]
=> [[1,2],[3]]
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> 3
[4]
=> [[1,2,3,4]]
=> 1
[3,1]
=> [[1,2,3],[4]]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[5]
=> [[1,2,3,4,5]]
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> 2
[3,2]
=> [[1,2,3],[4,5]]
=> 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> 2
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> 2
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 6
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8
Description
The row containing the largest entry of a standard tableau.
Mp00042: Integer partitions initial tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
[6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> 1 = 2 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 2 = 3 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 7 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 8 - 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Mp00202: Integer partitions first row removalInteger partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 0 = 1 - 1
[2]
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> 1 = 2 - 1
[3]
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [1,1]
=> 2 = 3 - 1
[4]
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> 1 = 2 - 1
[2,2]
=> [2]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[5]
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> 1 = 2 - 1
[3,2]
=> [2]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[6]
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> 1 = 2 - 1
[4,2]
=> [2]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,3]
=> [3]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 7 = 8 - 1
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Mp00044: Integer partitions conjugateInteger partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0 = 1 - 1
[2]
=> [1,1]
=> 0 = 1 - 1
[1,1]
=> [2]
=> 1 = 2 - 1
[3]
=> [1,1,1]
=> 0 = 1 - 1
[2,1]
=> [2,1]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> 2 = 3 - 1
[4]
=> [1,1,1,1]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,2]
=> [2,2]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 2 = 3 - 1
[1,1,1,1]
=> [4]
=> 3 = 4 - 1
[5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[3,2]
=> [2,2,1]
=> 1 = 2 - 1
[3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [5]
=> 4 = 5 - 1
[6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
[4,1,1]
=> [3,1,1,1]
=> 2 = 3 - 1
[3,3]
=> [2,2,2]
=> 1 = 2 - 1
[3,1,1,1]
=> [4,1,1]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [5,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [6]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [6,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [7]
=> 6 = 7 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> 7 = 8 - 1
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 $4+3+1 = 8$.
Mp00044: Integer partitions conjugateInteger partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0 = 1 - 1
[2]
=> [1,1]
=> 0 = 1 - 1
[1,1]
=> [2]
=> 1 = 2 - 1
[3]
=> [1,1,1]
=> 0 = 1 - 1
[2,1]
=> [2,1]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> 2 = 3 - 1
[4]
=> [1,1,1,1]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,2]
=> [2,2]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 2 = 3 - 1
[1,1,1,1]
=> [4]
=> 3 = 4 - 1
[5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[3,2]
=> [2,2,1]
=> 1 = 2 - 1
[3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [5]
=> 4 = 5 - 1
[6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
[4,1,1]
=> [3,1,1,1]
=> 2 = 3 - 1
[3,3]
=> [2,2,2]
=> 1 = 2 - 1
[3,1,1,1]
=> [4,1,1]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [5,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [6]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [6,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [7]
=> 6 = 7 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> 7 = 8 - 1
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 $4+3+1+0 = 8$.
Mp00317: Integer partitions odd partsBinary words
St000519: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 = 1 - 1
[2]
=> 0 => 0 = 1 - 1
[1,1]
=> 11 => 1 = 2 - 1
[3]
=> 1 => 0 = 1 - 1
[2,1]
=> 01 => 1 = 2 - 1
[1,1,1]
=> 111 => 2 = 3 - 1
[4]
=> 0 => 0 = 1 - 1
[3,1]
=> 11 => 1 = 2 - 1
[2,2]
=> 00 => 1 = 2 - 1
[2,1,1]
=> 011 => 2 = 3 - 1
[1,1,1,1]
=> 1111 => 3 = 4 - 1
[5]
=> 1 => 0 = 1 - 1
[4,1]
=> 01 => 1 = 2 - 1
[3,2]
=> 10 => 1 = 2 - 1
[3,1,1]
=> 111 => 2 = 3 - 1
[2,1,1,1]
=> 0111 => 3 = 4 - 1
[1,1,1,1,1]
=> 11111 => 4 = 5 - 1
[6]
=> 0 => 0 = 1 - 1
[5,1]
=> 11 => 1 = 2 - 1
[4,2]
=> 00 => 1 = 2 - 1
[4,1,1]
=> 011 => 2 = 3 - 1
[3,3]
=> 11 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 3 = 4 - 1
[2,1,1,1,1]
=> 01111 => 4 = 5 - 1
[1,1,1,1,1,1]
=> 111111 => 5 = 6 - 1
[2,1,1,1,1,1]
=> 011111 => 5 = 6 - 1
[1,1,1,1,1,1,1]
=> 1111111 => 6 = 7 - 1
[1,1,1,1,1,1,1,1]
=> 11111111 => 7 = 8 - 1
Description
The largest length of a factor maximising the subword complexity. Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal: $$ H_w = \max\{n: p_w(n)\text{ is maximal}\} $$ A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, [[St000294]].
The following 656 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000548The number of different non-empty partial sums of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. 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. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000982The length of the longest constant subword. St001267The length of the Lyndon factorization of the binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001814The number of partitions interlacing the given partition. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000996The number of exclusive left-to-right maxima of a permutation. St001777The number of weak descents in an integer composition. St000806The semiperimeter of the associated bargraph. St000012The area of a Dyck path. St000013The height of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000110The number of permutations less than or equal to a permutation in left weak order. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000167The number of leaves of an ordered tree. St000169The cocharge of a standard tableau. St000228The size of a partition. St000245The number of ascents of a permutation. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000479The Ramsey number of a graph. St000505The biggest entry in the block containing the 1. St000527The width of the poset. St000691The number of changes of a binary word. St000738The first entry in the last row of a standard tableau. St000808The number of up steps of the associated bargraph. St000820The number of compositions obtained by rotating the composition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000877The depth of the binary word interpreted as a path. St000909The number of maximal chains of maximal size in a poset. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001058The breadth of the ordered tree. St001330The hat guessing number of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001485The modular major index of a binary word. St001733The number of weak left to right maxima of a Dyck path. 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. St001933The largest multiplicity of a part in an integer partition. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000074The number of special entries. St000148The number of odd parts of a partition. St000203The number of external nodes of a binary tree. St000326The position of the first one in a binary word after appending a 1 at the end. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000356The number of occurrences of the pattern 13-2. St000374The number of exclusive right-to-left minima of a permutation. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000839The largest opener of a set partition. St000921The number of internal inversions of a binary word. 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. St001484The number of singletons of an integer partition. St001675The number of parts equal to the part in the reversed composition. St000668The least common multiple of the parts of the partition. St000058The order of a permutation. St000105The number of blocks in the set partition. St000678The number of up steps after the last double rise of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module 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: St000053The number of valleys of the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000172The Grundy number of a graph. St000308The height of the tree associated to a permutation. St000363The number of minimal vertex covers of a graph. St000383The last part of an integer composition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000504The cardinality of the first block of a set partition. St000528The height of a poset. St000636The hull number of a graph. St000708The product of the parts of an integer partition. St000722The number of different neighbourhoods in a graph. St000740The last entry of a permutation. St000766The number of inversions of an integer composition. St000823The number of unsplittable factors of the set partition. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000912The number of maximal antichains in a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000993The multiplicity of the largest part of an integer partition. 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. St001050The number of terminal closers of a set partition. St001062The maximal size of a block of a set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number 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. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001399The distinguishing number of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001697The shifted natural comajor index of a standard Young tableau. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000094The depth of an ordered tree. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000503The maximal difference between two elements in a common block. St000521The number of distinct subtrees of an ordered tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000674The number of hills of a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. 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 $c_{n-1}$. St001067The number of simple modules of dominant dimension at least two 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. 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. St000006The dinv of a Dyck path. St000925The number of topologically connected components of a set partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000306The bounce count of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000004The major index of a permutation. St000304The load of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000446The disorder of a permutation. St000874The position of the last double rise in a Dyck path. St000984The number of boxes below precisely one peak. St001298The number of repeated entries in the Lehmer code of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001427The number of descents of a signed permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001726The number of visible inversions of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St000809The reduced reflection length of the permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000291The number of descents of a binary word. St000314The number of left-to-right-maxima of a permutation. 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. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001717The largest size of an interval in a poset. St001963The tree-depth of a graph. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St000091The descent variation of a composition. St000154The sum of the descent bottoms of a permutation. St000168The number of internal nodes of an ordered tree. St000209Maximum difference of elements in cycles. St000238The number of indices that are not small weak excedances. St000292The number of ascents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000339The maf index of a permutation. St000653The last descent of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000989The number of final rises of a permutation. St001153The number of blocks with even minimum in a set partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. 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. St001671Haglund's hag of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000216The absolute length of a permutation. 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. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000061The number of nodes on the left branch of a binary tree. St000833The comajor index of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000990The first ascent of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000133The "bounce" of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001480The number of simple summands of the module J^2/J^3. St001727The number of invisible inversions of a permutation. St001812The biclique partition number of a graph. St001497The position of the largest weak excedence of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000204The number of internal nodes of a binary tree. St000224The sorting index of a permutation. St000305The inverse major index of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000144The pyramid weight of the Dyck path. St000159The number of distinct parts of the integer partition. St000235The number of indices that are not cyclical small weak excedances. St000454The largest eigenvalue of a graph if it is integral. St000692Babson and Steingrímsson's statistic of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000026The position of the first return of a 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. St000296The length of the symmetric border of a binary word. St000532The total number of rook placements on a Ferrers board. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000906The length of the shortest maximal chain in a poset. St000922The minimal number such that all substrings of this length are unique. 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. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000643The size of the largest orbit of antichains under Panyushev complementation. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001782The order of rowmotion on the set of order ideals of a poset. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000626The minimal period of a binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St000008The major index of the composition. St000047The number of standard immaculate tableaux of a given shape. St000161The sum of the sizes of the right subtrees of a binary tree. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000617The number of global maxima of a Dyck path. St000657The smallest part of an integer composition. St000682The Grundy value of Welter's game on a binary word. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000041The number of nestings of a perfect matching. St000145The Dyson rank of a partition. St000369The dinv deficit of a Dyck path. 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. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. 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. St000992The alternating sum of the parts of an integer partition. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001161The major index north count of a Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001759The Rajchgot index of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000014The number of parking functions supported by a Dyck path. St000255The number of reduced Kogan faces with the permutation as type. St000501The size of the first part in the decomposition of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with 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. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St000005The bounce statistic of a Dyck path. St000057The Shynar inversion number of a standard tableau. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000210Minimum over maximum difference of elements in cycles. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000931The number of occurrences of the pattern UUU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. 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. 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. St001172The number of 1-rises at odd height of a Dyck path. St001910The height of the middle non-run of a Dyck path. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000082The number of elements smaller than a binary tree in Tamari order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000060The greater neighbor of the maximum. St000654The first descent of a permutation. St000794The mak of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000117The number of centered tunnels of a Dyck path. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000424The number of occurrences of the pattern 132 or of the pattern 231 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000710The number of big deficiencies of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. 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. St001152The number of pairs with even minimum in a perfect matching. 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. 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. St001377The major index minus the number of inversions of a permutation. St001684The reduced word complexity of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000840The number of closers smaller than the largest opener in a perfect matching. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001555The order of a signed permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. 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. St000317The cycle descent number of a permutation. St000719The number of alignments in a perfect matching. St001108The 2-dynamic chromatic number of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St000225Difference between largest and smallest parts in a partition. St001746The coalition number of a graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001962The proper pathwidth of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000516The number of stretching pairs of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000770The major index of an integer partition when read from bottom to top. St000181The number of connected components of the Hasse diagram for the poset. St000420The number of Dyck paths that are weakly above a Dyck path. St000729The minimal arc length of a set partition. St001432The order dimension of the partition. St001808The box weight or horizontal decoration of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001644The dimension of a graph. St000487The length of the shortest cycle of a permutation. St001346The number of parking functions that give the same permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000045The number of linear extensions of a binary tree. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000438The position of the last up step in a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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. St001414Half the length of the longest odd length palindromic prefix 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. St001658The total number of rook placements on a Ferrers board. St001712The number of natural descents of a standard Young tableau. St000531The leading coefficient of the rook polynomial of an integer partition. St000878The number of ones minus the number of zeros of a binary word. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001523The degree of symmetry of a Dyck path. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001019Sum of the projective dimensions of the simple modules 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear 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. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001896The number of right descents of a signed permutations. 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. 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. St001008Number of indecomposable injective modules with projective dimension 1 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001863The number of weak excedances of a signed permutation. St001946The number of descents in a parking function. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). 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. St001115The number of even descents of a permutation. St001394The genus of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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 $eA$. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000477The weight of a partition according to Alladi. St000744The length of the path to the largest entry in a standard Young tableau. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000491The number of inversions of a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. 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. St000779The tier of a permutation. 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 $eA$. St000075The orbit size of a standard tableau under promotion. 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. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000762The sum of the positions of the weak records of an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000455The second largest eigenvalue of a graph if it is integral. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000464The Schultz index of a connected graph. St000736The last entry in the first row of a semistandard tableau. St001545The second Elser number of a connected graph. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation.