- FindStat

Your data matches 688 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010
(load all 12 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => []
 => 0
[1,0,1,0]
 => [1]
 => 1
[1,1,0,0]
 => []
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => 2
[1,1,0,0,1,0]
 => [2]
 => 1
[1,1,0,1,0,0]
 => [1]
 => 1
[1,1,1,0,0,0]
 => []
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,4,4]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [3,3,3]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => 3

Description

The length of the partition.

Matching statistic: St000012
(load all 15 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => 3

Description

The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences'''

$(a_1,\ldots,a_n)$ such that

$a_1 = 0, a_{k+1} \leq a_k + 1$ . 2. The generating function

$\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence

$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of

$q,t$ -Catalan numbers.

Matching statistic: St000160
(load all 6 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => []
 => 0
[1,0,1,0]
 => [1]
 => 1
[1,1,0,0]
 => []
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => 2
[1,1,0,0,1,0]
 => [2]
 => 1
[1,1,0,1,0,0]
 => [1]
 => 1
[1,1,1,0,0,0]
 => []
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,4,4]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [3,3,3]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => 3

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].

Matching statistic: St000394
(load all 32 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => 3

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St000548
(load all 6 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => []
 => 0
[1,0,1,0]
 => [1]
 => 1
[1,1,0,0]
 => []
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => 2
[1,1,0,0,1,0]
 => [2]
 => 1
[1,1,0,1,0,0]
 => [1]
 => 1
[1,1,1,0,0,0]
 => []
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,4,4]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [3,3,3]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => 3

Description

The number of different non-empty partial sums of an integer partition.

Matching statistic: St000032
(load all 15 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 6 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1

Description

The number of elements smaller than the given Dyck path in the Tamari Order.

Matching statistic: St000008
(load all 3 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,1] => [1,1] => 1
[1,1,0,0]
 => [2] => [2] => 0
[1,0,1,1,0,0]
 => [1,2] => [2,1] => 2
[1,1,0,0,1,0]
 => [2,1] => [1,2] => 1
[1,1,0,1,0,0]
 => [2,1] => [1,2] => 1
[1,1,1,0,0,0]
 => [3] => [3] => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1] => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2] => 2
[1,1,0,1,1,0,0,0]
 => [2,2] => [2,2] => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,3] => 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,3] => 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,3] => 1
[1,1,1,1,0,0,0,0]
 => [4] => [4] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2] => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [2,3] => 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [2,3] => 2
[1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [2,3] => 2
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,4] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,4] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,4] => 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,4] => 1
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [5] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => [5,1] => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4] => [4,2] => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [4,2] => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3] => [3,3] => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [3,3] => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [3,3] => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,2] => [2,4] => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,2] => [2,4] => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,2] => [2,4] => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,2] => [2,4] => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => [1,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1] => [1,5] => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,1] => [1,5] => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1] => [1,5] => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,1] => [1,5] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6] => [6] => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => [6,1] => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,5] => [5,2] => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,5] => [5,2] => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,4] => [4,3] => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,4] => [4,3] => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4] => [4,3] => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3] => [3,4] => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [4,3] => [3,4] => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [4,3] => [3,4] => 3

Description

The major index of the composition. The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

Matching statistic: St000009
(load all 3 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [[1]]
 => 0
[1,0,1,0]
 => [1,2] => [[1,2]]
 => 1
[1,1,0,0]
 => [2,1] => [[1],[2]]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [[1],[2],[3]]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[1,3],[2],[4]]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [[1,4],[2],[3]]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [[1,4],[2],[3]]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [[1,4],[2,5],[3]]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [[1,4],[2],[3],[5]]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [[1,5],[2],[3],[4]]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [[1,5],[2],[3],[4]]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [[1,5],[2],[3],[4]]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3] => [[1,3],[2,4],[5],[6]]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => [[1,3],[2],[4],[5],[6]]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,1,6,5,4] => [[1,4],[2,5],[3,6]]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => [[1,4],[2,5],[3],[6]]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,5,4,2,1] => [[1,4],[2],[3],[5],[6]]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => [[1,5],[2,6],[3],[4]]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => [[1,5],[2,6],[3],[4]]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,3,6,5,2,1] => [[1,5],[2,6],[3],[4]]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,6,5,3,2,1] => [[1,5],[2],[3],[4],[6]]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => [[1,6],[2],[3],[4],[5]]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,4,3,6,2,1] => [[1,6],[2],[3],[4],[5]]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,4,6,3,2,1] => [[1,6],[2],[3],[4],[5]]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5]]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7]]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,6,5,4,3] => [[1,3],[2,4],[5],[6],[7]]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,6,5,4,3,1] => [[1,3],[2],[4],[5],[6],[7]]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => [[1,4],[2,5],[3,6],[7]]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,2,7,6,5,4,1] => [[1,4],[2,5],[3],[6],[7]]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,7,6,5,4,2,1] => [[1,4],[2],[3],[5],[6],[7]]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => [[1,5],[2,6],[3,7],[4]]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [4,3,2,7,6,5,1] => [[1,5],[2,6],[3,7],[4]]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [4,3,7,6,5,2,1] => [[1,5],[2,6],[3],[4],[7]]
 => 3

Description

The charge of a standard tableau.

Matching statistic: St000053
(load all 15 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 3

Description

The number of valleys of the Dyck path.

Matching statistic: St000147
(load all 11 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => []
 => []
 => 0
[1,0,1,0]
 => [1]
 => [1]
 => 1
[1,1,0,0]
 => []
 => []
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [2]
 => 2
[1,1,0,0,1,0]
 => [2]
 => [1,1]
 => 1
[1,1,0,1,0,0]
 => [1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => []
 => []
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [3]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2,2]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [2]
 => 2
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [4]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [3,3]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [3]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [2,2,2]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2,2]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [2]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [5]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [4,4]
 => 4
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [4]
 => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [3,3,3]
 => 3
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [3,3]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [3]
 => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [2,2,2,2]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [2,2,2]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [2,2]
 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [2]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [1,1,1,1,1]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [1,1,1,1]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,1,1]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [6]
 => 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => [5,5]
 => 5
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [5]
 => 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3]
 => [4,4,4]
 => 4
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2]
 => [4,4]
 => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [4]
 => 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,4,4]
 => [3,3,3,3]
 => 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [3,3,3]
 => [3,3,3]
 => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => [3,3]
 => 3

Description

The largest part of an integer partition.

The following 678 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000169The cocharge of a standard tableau. St000211The rank of the set partition. St000330The (standard) major index of a standard tableau. St000378The diagonal inversion number of an integer partition. St000984The number of boxes below precisely one peak. St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001161The major index north count of a Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001697The shifted natural comajor index of a standard Young tableau. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000058The order of a permutation. St000381The largest part of an integer composition. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000738The first entry in the last row of a standard tableau. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St000971The smallest closer of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001814The number of partitions interlacing the given partition. St000439The position of the first down step of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000081The number of edges of a graph. St000157The number of descents of a standard tableau. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000445The number of rises of length 1 of a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000846The maximal number of elements covering an element of a poset. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001120The length of a longest path in a graph. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001280The number of parts of an integer partition that are at least two. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001479The number of bridges of a graph. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001512The minimum rank of a graph. St001721The degree of a binary word. St001759The Rajchgot index of a permutation. St001777The number of weak descents in an integer composition. St001826The maximal number of leaves on a vertex of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000013The height of a Dyck path. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000382The first part of an integer composition. St000383The last part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000733The row containing the largest entry of a standard tableau. 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. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. 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. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. 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. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. St000521The number of distinct subtrees of an ordered tree. 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. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000947The major index east count of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000391The sum of the positions of the ones in a binary word. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000503The maximal difference between two elements in a common block. St000693The modular (standard) major index of a standard tableau. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000392The length of the longest run of ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000446The disorder of a permutation. St000490The intertwining number of a set partition. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St001372The length of a longest cyclic run of ones of a binary word. 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. St001726The number of visible inversions of a permutation. St000054The first entry of the permutation. St000069The number of maximal elements of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000708The product of the parts of an integer partition. St000925The number of topologically connected components of a set partition. St001062The maximal size of a block of a set partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001808The box weight or horizontal decoration of a Dyck path. St000007The number of saliances of the permutation. St000507The number of ascents of a standard tableau. St000662The staircase size of the code of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000809The reduced reflection length of the permutation. St000833The comajor index of a permutation. St000845The maximal number of elements covered by an element in a poset. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001695The natural comajor index of a standard Young tableau. St000485The length of the longest cycle of a permutation. St000527The width of the poset. St000528The height of a poset. St000844The size of the largest block in the direct sum decomposition of a permutation. St000883The number of longest increasing subsequences of a permutation. St000912The number of maximal antichains in a poset. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001343The dimension of the reduced incidence algebra of a poset. St000148The number of odd parts of a partition. St000632The jump number of the poset. St000783The side length of the largest staircase partition fitting into a partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000071The number of maximal chains in a poset. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001717The largest size of an interval in a poset. St000306The bounce count of a Dyck path. St000004The major index of a permutation. St000546The number of global descents of a permutation. St001671Haglund's hag of a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000643The size of the largest orbit of antichains under Panyushev complementation. St001933The largest multiplicity of a part in an integer partition. St000296The length of the symmetric border of a binary word. St000393The number of strictly increasing runs in a binary word. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000797The stat`` of a permutation. St000798The makl of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001267The length of the Lyndon factorization of the binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001884The number of borders of a binary word. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000015The number of peaks of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000617The number of global maxima of a Dyck path. St000626The minimal period of a binary word. St000653The last descent of a permutation. St000657The smallest part of an integer composition. St000682The Grundy value of Welter's game on a binary word. St000794The mak of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000983The length of the longest alternating subword. St000989The number of final rises of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001118The acyclic chromatic index of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001313The number of Dyck paths above the lattice path given by a binary word. St001489The maximum of the number of descents and the number of inverse descents. St000031The number of cycles in the cycle decomposition of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000203The number of external nodes of a binary tree. St000204The number of internal nodes of a binary tree. St000308The height of the tree associated to a permutation. St000331The number of upper interactions of a Dyck path. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000470The number of runs in a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (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. St000674The number of hills of a Dyck path. St000691The number of changes of a binary word. St000921The number of internal inversions of a binary word. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001910The height of the middle non-run of a Dyck path. St000806The semiperimeter of the associated bargraph. St000161The sum of the sizes of the right subtrees of a binary tree. St000159The number of distinct parts of the integer partition. St000245The number of ascents of a permutation. St000354The number of recoils of a permutation. St000651The maximal size of a rise in a permutation. St000770The major index of an integer partition when read from bottom to top. St000829The Ulam distance of a permutation to the identity permutation. St000993The multiplicity of the largest part of an integer partition. St000225Difference between largest and smallest parts in a partition. St000355The number of occurrences of the pattern 21-3. 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. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St001571The Cartan determinant of the integer partition. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001675The number of parts equal to the part in the reversed composition. St000041The number of nestings of a perfect matching. St000795The mad of a permutation. St000237The number of small exceedances. St000539The number of odd inversions of a permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000214The number of adjacencies of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000153The number of adjacent cycles of a permutation. St000100The number of linear extensions of a poset. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000304The load of a permutation. St000314The number of left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000654The first descent of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001128The exponens consonantiae of a partition. 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

$A$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001360The number of covering relations in Young's lattice below a partition. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001497The position of the largest weak excedence of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000082The number of elements smaller than a binary tree in Tamari order. St000117The number of centered tunnels of a Dyck path. St000216The absolute length of a permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000338The number of pixed points of a permutation. St000740The last entry of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001152The number of pairs with even minimum in a perfect matching. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001239The largest vector space dimension of the double dual of a simple module 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. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000246The number of non-inversions of a permutation. St000356The number of occurrences of the pattern 13-2. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000702The number of weak deficiencies of a permutation. St001432The order dimension of the partition. St000463The number of admissible inversions of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000006The dinv of a Dyck path. St000223The number of nestings in the permutation. St000325The width of the tree associated to a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000542The number of left-to-right-minima of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000292The number of ascents of a binary word. St000648The number of 2-excedences of a permutation. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000843The decomposition number of a perfect matching. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001397Number of pairs of incomparable elements in a finite poset. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000864The number of circled entries of the shifted recording tableau of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000002The number of occurrences of the pattern 123 in a permutation. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000057The Shynar inversion number of a standard tableau. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000224The sorting index of a permutation. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000796The stat' 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. 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

$eA$ . 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. St000240The number of indices that are not small excedances. St000991The number of right-to-left minima of a permutation. St001530The depth of a Dyck path. St000042The number of crossings of a perfect matching. St000080The rank of the poset. St000091The descent variation of a composition. St000168The number of internal nodes of an ordered tree. St000233The number of nestings of a set partition. St000238The number of indices that are not small weak excedances. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. 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. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 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. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001077The prefix exchange distance of a permutation. St001094The depth index of a set partition. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001153The number of blocks with even minimum in a set partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001428The number of B-inversions of a signed permutation. St001649The length of a longest trail in a graph. St001718The number of non-empty open intervals in a poset. St001842The major index of a set partition. St001843The Z-index of a set partition. St001869The maximum cut size of a graph. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000086The number of subgraphs. St000166The depth minus 1 of an ordered tree. St000239The number of small weak excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St000990The first ascent of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001187The number of simple modules with grade at least one in the corresponding Nakayama 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000094The depth of an ordered tree. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000472The sum of the ascent bottoms of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001346The number of parking functions that give the same permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000061The number of nodes on the left branch of a binary tree. 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. St001959The product of the heights of the peaks of a Dyck path. St000462The major index minus the number of excedences of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000652The maximal difference between successive positions of a permutation. St001427The number of descents of a signed permutation. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000367The number of simsun double descents of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001411The number of patterns 321 or 3412 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000619The number of cyclic descents of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000961The shifted major index of a permutation. St001330The hat guessing number of a graph. St000647The number of big descents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000357The number of occurrences of the pattern 12-3. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St000746The number of pairs with odd minimum in a perfect matching. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St001396Number of triples of incomparable elements in a finite poset. St000909The number of maximal chains of maximal size in a poset. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St001377The major index minus the number of inversions of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001684The reduced word complexity of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000241The number of cyclical small excedances. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000376The bounce deficit of a Dyck path. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000461The rix statistic of a permutation. St000719The number of alignments in a perfect matching. St000732The number of double deficiencies of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001082The number of boxed occurrences of 123 in a permutation. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001434The number of negative sum pairs of a signed permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001552The number of inversions between excedances and fixed points of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001727The number of invisible inversions of a permutation. St000060The greater neighbor of the maximum. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001220The width of a permutation. St001555The order of a signed permutation. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000327The number of cover relations in a poset. St001668The number of points of the poset minus the width of the poset. St000456The monochromatic index of a connected graph. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. 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

$eA$ . St001863The number of weak excedances of a signed permutation. St001209The pmaj statistic of a parking function. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001513The number of nested exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001712The number of natural descents of a standard Young tableau. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000115The single entry in the last row. St000942The number of critical left to right maxima of the parking functions. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001937The size of the center of a parking function. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000834The number of right outer peaks of a permutation. St001438The number of missing boxes of a skew partition. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001060The distinguishing index of a graph. St001394The genus of a permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000731The number of double exceedences of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000562The number of internal points of a set partition. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000173The segment statistic of a semistandard tableau. St000353The number of inner valleys of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000761The number of ascents in an integer composition. St000836The number of descents of distance 2 of a permutation. St001114The number of odd descents of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001403The number of vertical separators in a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001689The number of celebrities in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001862The number of crossings of a signed permutation. St000089The absolute variation of a composition. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000307The number of rowmotion orbits of a poset. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000988The orbit size of a permutation under Foata's bijection. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001928The number of non-overlapping descents in a permutation. St000422The energy of a graph, if it is integral.