- 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: St000021
(load all 118 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000035
(load all 299 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left outer peaks of a permutation. A left 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

$w_1 > w_2$ . In other words, it is a peak in the word

$[0,w_1,..., w_n]$ . This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

Matching statistic: St000337
(load all 68 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$ where

$\textrm{inv} \, \tau_{i}$ is the number of inversions of

$\tau_{i}$ .

Matching statistic: St000374
(load all 68 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000703
(load all 56 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of deficiencies of a permutation. This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$ The number of exceedances is [[St000155]].

Matching statistic: St000834
(load all 248 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of right outer peaks of a permutation. A right 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

$n$ if

$w_n > w_{n-1}$ . In other words, it is a peak in the word

$[w_1,..., w_n,0]$ .

Matching statistic: St000884
(load all 103 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of isolated descents of a permutation. A descent

$i$ is isolated if neither

$i+1$ nor

$i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].

Matching statistic: St000994
(load all 209 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation

$\pi$ is an index

$i$ such that

$\pi^{-1}(i) < i > \pi(i)$ . Analogously, a '''cycle valley''' is an index

$i$ such that

$\pi^{-1}(i) > i < \pi(i)$ . Clearly, every cycle of

$\pi$ contains as many peaks as valleys.

Matching statistic: St000201
(load all 11 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 3
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [[[[[.,.],[.,.]],.],.],.]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[[[.,.],[[.,.],.]],.],.]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[[[.,.],.],.]],.]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => 1
[]
 => [1,0]
 => [.,.]
 => 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: St000321
(load all 3 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of integer partitions of n that are dominated by an integer partition. A partition

$\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition

$\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if

$\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all

$k$ .

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

St000345The number of refinements of a partition. St000676The number of odd rises of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. 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. St000935The number of ordered refinements of an integer partition. 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: St001597The Frobenius rank of a skew partition. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000336The leg major index of a standard tableau. St000386The number of factors DDU in a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001394The genus of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000105The number of blocks in the set partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000528The height of a poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001343The dimension of the reduced incidence algebra of a poset. St001471The magnitude of a Dyck path. St001717The largest size of an interval in a poset. St001732The number of peaks visible from the left. St000004The major index of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000141The maximum drop size of a permutation. 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. St000159The number of distinct parts of the integer partition. St000169The cocharge of a standard tableau. St000257The number of distinct parts of a partition that occur at least twice. St000330The (standard) major index of a standard tableau. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000377The dinv defect of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St000920The logarithmic height of a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001665The number of pure excedances of a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001928The number of non-overlapping descents in a permutation. 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). St000325The width of the tree associated to a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group 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. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001261The Castelnuovo-Mumford regularity of a graph. St001389The number of partitions of the same length below the given integer partition. St000023The number of inner peaks of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000291The number of descents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000670The reversal length of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001280The number of parts of an integer partition that are at least two. St001298The number of repeated entries in the Lehmer code of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001427The number of descents of a signed permutation. St001657The number of twos in an integer partition. St001726The number of visible inversions of a permutation. St001812The biclique partition number of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000062The length of the longest increasing subsequence of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000086The number of subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000258The burning number of a graph. St000273The domination number of a graph. St000288The number of ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000485The length of the longest cycle of a permutation. St000532The total number of rook placements on a Ferrers board. St000542The number of left-to-right-minima of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000738The first entry in the last row of a standard tableau. St000822The Hadwiger number of the graph. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. 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. St001116The game chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001339The irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001400The total number of Littlewood-Richardson tableaux of given shape. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001725The harmonious chromatic number of a graph. St001734The lettericity of a graph. St001814The number of partitions interlacing the given partition. St001963The tree-depth of a graph. St000080The rank of the poset. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000228The size of a partition. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000358The number of occurrences of the pattern 31-2. St000361The second Zagreb index of a graph. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000459The hook length of the base cell of a partition. St000521The number of distinct subtrees of an ordered tree. St000523The number of 2-protected nodes of a rooted tree. St000536The pathwidth of a graph. St000539The number of odd inversions of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000778The metric dimension of a graph. St000779The tier of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000787The number of flips required to make a perfect matching noncrossing. St000829The Ulam distance of a permutation to the identity permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000919The number of maximal left branches of a binary tree. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. 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. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001153The number of blocks with even minimum in a set partition. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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 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. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001358The largest degree of a regular subgraph of a graph. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001479The number of bridges of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 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. St001697The shifted natural comajor index of a standard Young tableau. St001792The arboricity of a graph. St001869The maximum cut size of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000702The number of weak deficiencies of a permutation. St000216The absolute length of a permutation. St000389The number of runs of ones of odd length in a binary word. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000640The rank of the largest boolean interval in a poset. St000809The reduced reflection length of the permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001152The number of pairs with even minimum in a perfect matching. St001372The length of a longest cyclic run of ones of a binary word. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St000668The least common multiple of the parts of the partition. St000746The number of pairs with odd minimum in a perfect matching. St000925The number of topologically connected components of a set partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001959The product of the heights of the peaks of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000253The crossing number of a set partition. St000392The length of the longest run of ones in a binary word. St000493The los statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000653The last descent of a permutation. St000693The modular (standard) major index of a standard tableau. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000794The mak of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001592The maximal number of simple paths between any two different vertices of a graph. St001896The number of right descents of a signed permutations. St001668The number of points of the poset minus the width of the poset. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000862The number of parts of the shifted shape of a permutation. St000534The number of 2-rises of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St001470The cyclic holeyness of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. 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

$eA$ . 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

$eA$ . St000781The number of proper colouring schemes of a Ferrers diagram. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001823The Stasinski-Voll length 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. St001175The size of a partition minus the hook length of the base cell. St000630The length of the shortest palindromic decomposition of a binary word. St000805The number of peaks of the associated bargraph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000003The number of standard Young tableaux of the partition. St000100The number of linear extensions of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000460The hook length of the last cell along the main diagonal of an integer partition. St000633The size of the automorphism group of a poset. St000655The length of the minimal rise of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000768The number of peaks in an integer composition. St000806The semiperimeter of the associated bargraph. St000807The sum of the heights of the valleys of the associated bargraph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000910The number of maximal chains of minimal length in a poset. 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. St001009Number of indecomposable injective modules with projective dimension g when g is 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. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. 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

$g$ . 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. St001335The cardinality of a minimal cycle-isolating set of a graph. 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. St001432The order dimension of the partition. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs 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. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001712The number of natural descents of a standard Young tableau. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000017The number of inversions of a standard tableau. St000145The Dyson rank of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001584The area statistic between a Dyck path and its bounce path. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001487The number of inner corners of a skew partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St000307The number of rowmotion orbits of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St001624The breadth of a lattice. St001490The number of connected components of a skew partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000396The register function (or Horton-Strahler number) of a binary tree. St001128The exponens consonantiae of a partition. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001115The number of even descents of a permutation. St001060The distinguishing index of a graph. St000260The radius of a connected graph. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000891The number of distinct diagonal sums of a permutation matrix. St001780The order of promotion on the set of standard tableaux of given shape. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001569The maximal modular displacement of a permutation. St001645The pebbling number of a connected graph. St001863The number of weak excedances of a signed permutation. St000259The diameter of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000741The Colin de Verdière graph invariant. St001423The number of distinct cubes in a binary word. St001862The number of crossings of a signed permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 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

$K[x]/(x^n)$ . St000886The number of permutations with the same antidiagonal sums. St001111The weak 2-dynamic chromatic number 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. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000538The number of even inversions of a permutation. St000563The number of overlapping pairs of blocks of a set partition. St000649The number of 3-excedences of a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000872The number of very big descents of a permutation. 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. St001093The detour number of a graph. St001566The length of the longest arithmetic progression in a permutation. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001644The dimension of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001638The book thickness of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000477The weight of a partition according to Alladi. 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. 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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001545The second Elser number of a connected graph. St000284The Plancherel distribution on integer partitions. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer 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. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000079The number of alternating sign matrices for a given Dyck path. St000315The number of isolated vertices of a graph. St000909The number of maximal chains of maximal size in a poset. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001413Half the length of the longest even length palindromic prefix of a binary word. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000365The number of double ascents of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in a permutation. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001857The number of edges in the reduced word graph of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000474Dyson's crank of a partition. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001118The acyclic chromatic index of a graph. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000464The Schultz index of a connected graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in 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

$K[x]/(x^n)$ . 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

$K[x]/(x^n)$ . St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000739The first entry in the last row of a semistandard tableau. St000879The number of long braid edges in the graph of braid moves of a permutation. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001684The reduced word complexity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation.