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Your data matches 709 different statistics following compositions of up to 3 maps.
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Matching statistic: St000183
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000183: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 1
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000920
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000920: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000920: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> 1
[1,1]
=> [1,1,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001597
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00179: Integer partitions āto skew partitionā¶ Skew partitions
St001597: Skew partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001597: Skew partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> 1
[2]
=> [[2],[]]
=> 1
[1,1]
=> [[1,1],[]]
=> 1
[3]
=> [[3],[]]
=> 1
[2,1]
=> [[2,1],[]]
=> 1
[1,1,1]
=> [[1,1,1],[]]
=> 1
[4]
=> [[4],[]]
=> 1
[3,1]
=> [[3,1],[]]
=> 1
[2,2]
=> [[2,2],[]]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[5]
=> [[5],[]]
=> 1
[4,1]
=> [[4,1],[]]
=> 1
[3,2]
=> [[3,2],[]]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> 1
[2,2,1]
=> [[2,2,1],[]]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[6]
=> [[6],[]]
=> 1
[5,1]
=> [[5,1],[]]
=> 1
[4,2]
=> [[4,2],[]]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> 1
[3,3]
=> [[3,3],[]]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> 2
[3,1,1,1]
=> [[3,1,1,1],[]]
=> 1
[2,2,2]
=> [[2,2,2],[]]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 1
Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St000480
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000480: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000480: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 0 = 1 - 1
[2]
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> 0 = 1 - 1
[3]
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> 0 = 1 - 1
[4]
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> 0 = 1 - 1
[2,2]
=> [2]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[5]
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> 0 = 1 - 1
[3,2]
=> [2]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[6]
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> 0 = 1 - 1
[4,2]
=> [2]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,3]
=> [3]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[2,2,2]
=> [2,2]
=> 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = 1 - 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000660
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000660: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000660: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000092
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St000092: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St000092: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[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] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[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] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[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] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1
[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] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[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] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 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] => 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000201
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00140: Dyck paths ālogarithmic height to pruning numberā¶ Binary trees
St000201: Binary trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00140: Dyck paths ālogarithmic height to pruning numberā¶ Binary trees
St000201: Binary trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [.,.]
=> 1
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000291
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00316: Binary words āinverse Foata bijectionā¶ Binary words
St000291: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00316: Binary words āinverse Foata bijectionā¶ Binary words
St000291: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 10 => 1
[2]
=> 100 => 010 => 1
[1,1]
=> 110 => 110 => 1
[3]
=> 1000 => 0010 => 1
[2,1]
=> 1010 => 0110 => 1
[1,1,1]
=> 1110 => 1110 => 1
[4]
=> 10000 => 00010 => 1
[3,1]
=> 10010 => 00110 => 1
[2,2]
=> 1100 => 1010 => 2
[2,1,1]
=> 10110 => 01110 => 1
[1,1,1,1]
=> 11110 => 11110 => 1
[5]
=> 100000 => 000010 => 1
[4,1]
=> 100010 => 000110 => 1
[3,2]
=> 10100 => 10010 => 2
[3,1,1]
=> 100110 => 001110 => 1
[2,2,1]
=> 11010 => 10110 => 2
[2,1,1,1]
=> 101110 => 011110 => 1
[1,1,1,1,1]
=> 111110 => 111110 => 1
[6]
=> 1000000 => 0000010 => 1
[5,1]
=> 1000010 => 0000110 => 1
[4,2]
=> 100100 => 100010 => 2
[4,1,1]
=> 1000110 => 0001110 => 1
[3,3]
=> 11000 => 01010 => 2
[3,2,1]
=> 101010 => 100110 => 2
[3,1,1,1]
=> 1001110 => 0011110 => 1
[2,2,2]
=> 11100 => 11010 => 2
[2,2,1,1]
=> 110110 => 101110 => 2
[2,1,1,1,1]
=> 1011110 => 0111110 => 1
[1,1,1,1,1,1]
=> 1111110 => 1111110 => 1
Description
The number of descents of a binary word.
Matching statistic: St000298
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00232: Dyck paths āparallelogram posetā¶ Posets
St000298: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00232: Dyck paths āparallelogram posetā¶ Posets
St000298: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> ([],1)
=> 1
[2]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[1,1]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1
[3]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00232: Dyck paths āparallelogram posetā¶ Posets
St000307: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00232: Dyck paths āparallelogram posetā¶ Posets
St000307: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> ([],1)
=> 1
[2]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[1,1]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1
[3]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
The following 699 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000352The Elizalde-Pak rank of a permutation. St000390The number of runs of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000527The width of the poset. St000758The length of the longest staircase fitting into an integer composition. St000862The number of parts of the shifted shape of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000481The number of upper covers of a partition in dominance order. St000632The jump number of the poset. St000871The number of very big ascents of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001394The genus of a permutation. St001471The magnitude of a Dyck path. St001728The number of invisible descents of a permutation. St000007The number of saliances of the permutation. St000035The number of left outer peaks of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000099The number of valleys of a permutation, including the boundary. St000185The weighted size of a partition. St000251The number of nonsingleton blocks of a set partition. St000292The number of ascents of a binary word. St000325The width of the tree associated to a permutation. St000336The leg major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000381The largest part of an integer composition. St000387The matching number of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000522The number of 1-protected nodes of a rooted tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000679The pruning number of an ordered tree. St000703The number of deficiencies of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000808The number of up steps of the associated bargraph. St000903The number of different parts of an integer composition. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001111The weak 2-dynamic chromatic number of a graph. St001176The size of a partition minus its first part. St001261The Castelnuovo-Mumford regularity of a graph. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001432The order dimension of the partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001486The number of corners of the ribbon associated with an integer composition. St001665The number of pure excedances of a permutation. St001716The 1-improper chromatic number of a graph. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001884The number of borders of a binary word. St000013The height of a Dyck path. St000017The number of inversions of a standard tableau. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000091The descent variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000397The Strahler number of a rooted tree. St000444The length of the maximal rise of a Dyck path. St000523The number of 2-protected nodes of a rooted tree. St000535The rank-width of a graph. St000552The number of cut vertices of a graph. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000664The number of right ropes of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000731The number of double exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000834The number of right outer peaks of a permutation. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000966Number of peaks minus the global dimension 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. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. 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. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001393The induced matching number of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001584The area statistic between a Dyck path and its bounce path. St001638The book thickness of a graph. 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. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001874Lusztig's a-function for the symmetric group. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001928The number of non-overlapping descents in a permutation. St000354The number of recoils of a permutation. St000640The rank of the largest boolean interval in a poset. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000539The number of odd inversions of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000886The number of permutations with the same antidiagonal sums. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001330The hat guessing number of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St000366The number of double descents of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000562The number of internal points of a set partition. St000565The major index of a set partition. 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. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000658The number of rises of length 2 of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001220The width of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000570The Edelman-Greene number of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001153The number of blocks with even minimum in a set partition. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001427The number of descents of a signed permutation. St001481The minimal height of a peak of a Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000317The cycle descent number of a permutation. St000335The difference of lower and upper interactions. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000443The number of long tunnels of a Dyck path. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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 $eA$. 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:
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 $eA$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001224Let X be the direct sum of all simple modules 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001411The number of patterns 321 or 3412 in a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001731The factorization defect of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000842The breadth of a permutation. St000993The multiplicity of the largest part of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001162The minimum jump of a permutation. St000223The number of nestings in the permutation. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001715The number of non-records in a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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 $K[x]/(x^n)$. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000542The number of left-to-right-minima of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000015The number of peaks of a Dyck path. St000054The first entry of the permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000184The size of the centralizer of any permutation of given cycle type. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000402Half the size of the symmetry class of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000630The length of the shortest palindromic decomposition of a binary word. St000654The first descent of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000759The smallest missing part in an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. 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. St001151The number of blocks with odd minimum. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama 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 $eA$. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001267The length of the Lyndon factorization of the binary word. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001517The length of a longest pair of twins in a permutation. St001530The depth of a 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. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001667The maximal size of a pair of weak twins for a permutation. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000061The number of nodes on the left branch of a binary tree. St000308The height of the tree associated to a permutation. St000367The number of simsun double descents of a permutation. St000370The genus of a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001550The number of inversions between exceedances where the greater exceedance is linked. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001856The number of edges in the reduced word graph of a permutation. St001964The interval resolution global dimension of a poset. St000159The number of distinct parts of the integer partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Mƶbius function of a poset. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001487The number of inner corners of a skew partition. St001568The smallest positive integer that does not appear twice in the partition. St001712The number of natural descents of a standard Young tableau. 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 $eA$. St000908The length of the shortest maximal antichain in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000297The number of leading ones in a binary word. St000629The defect of a binary word. St001128The exponens consonantiae of a partition. St000060The greater neighbor of the maximum. St000260The radius of a connected graph. St000914The sum of the values of the Mƶbius function of a poset. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001896The number of right descents of a signed permutations. St001720The minimal length of a chain of small intervals in a lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001490The number of connected components of a skew partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001780The order of promotion on the set of standard tableaux of given shape. St001881The number of factors of a lattice as a Cartesian product of lattices. 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. 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. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001616The number of neutral elements in a lattice. St001846The number of elements which do not have a complement in the lattice. St000454The largest eigenvalue of a graph if it is integral. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St001569The maximal modular displacement of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St000667The greatest common divisor of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000026The position of the first return of a Dyck path. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000160The multiplicity of the smallest part of a partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000734The last entry in the first row of a standard tableau. St000784The maximum of the length and the largest part of the integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001462The number of factors of a standard tableaux under concatenation. St001488The number of corners of a skew partition. St001527The cyclic permutation representation number of an integer partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000878The number of ones minus the number of zeros of a binary word. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001260The permanent of an alternating sign matrix. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001590The crossing number of a perfect matching. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001889The size of the connectivity set of a signed permutation. St001946The number of descents in a parking function. St000022The number of fixed points of a permutation. St000058The order of a permutation. St000153The number of adjacent cycles of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000392The length of the longest run of ones in a binary word. St000405The number of occurrences of the pattern 1324 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000807The sum of the heights of the valleys of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000894The trace of an alternating sign matrix. St000905The number of different multiplicities of parts of an integer composition. St000943The number of spots the most unlucky car had to go further in a parking function. St000982The length of the longest constant subword. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001845The number of join irreducibles minus the rank of a lattice. St001850The number of Hecke atoms of a permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000767The number of runs in an integer composition. St000891The number of distinct diagonal sums of a permutation matrix. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001566The length of the longest arithmetic progression in a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000618The number of self-evacuating tableaux of given shape. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St000455The second largest eigenvalue of a graph if it is integral. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000045The number of linear extensions of a binary tree. St000460The hook length of the last cell along the main diagonal of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements 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. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000939The number of characters of the symmetric group whose value on the partition is positive. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000735The last entry on the main diagonal of a standard tableau. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000729The minimal arc length of a set partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001806The upper middle entry of a permutation. St000039The number of crossings of a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000462The major index minus the number of excedences of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000504The cardinality of the first block of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000823The number of unsplittable factors of the set partition. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001537The number of cyclic crossings of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001705The number of occurrences of the pattern 2413 in a permutation. St001781The interlacing number of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001903The number of fixed points of a parking function. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001624The breadth of a lattice. St001875The number of simple modules with projective dimension at most 1. St000633The size of the automorphism group of a poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. 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. St001399The distinguishing number of a poset. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000850The number of 1/2-balanced pairs in a poset. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001851The number of Hecke atoms of a signed permutation. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000831The number of indices that are either descents or recoils. St001472The permanent of the Coxeter matrix of the poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000102The charge of a semistandard tableau. St001948The number of augmented double ascents of a permutation.
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