- FindStat

Your data matches 795 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000686
(load all 33 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000686: Dyck paths ⟶ 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]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[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]
 => 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]
 => 3
[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]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
[]
 => [1,0]
 => 1

Description

The finitistic dominant dimension of a Dyck path. To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.

Matching statistic: St001203
(load all 41 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001203: Dyck paths ⟶ 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]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[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]
 => 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]
 => 3
[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]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1
[]
 => [1,0]
 => 1

Description

We 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: In the list

$L$ delete the first entry

$c_0$ and substract from all other entries

$n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.

Matching statistic: St000021
(load all 130 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ 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]
 => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [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 334 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-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]
 => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => 1 = 2 - 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] => 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] => 2 = 3 - 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] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [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: St000306
(load all 41 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
[]
 => [1,0]
 => 0 = 1 - 1

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000834
(load all 260 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-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]
 => [1,2] => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => 1 = 2 - 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] => 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] => 2 = 3 - 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] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [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 117 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ 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]
 => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [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 208 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ 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]
 => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [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: St001197
(load all 41 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
[]
 => [1,0]
 => 0 = 1 - 1

Description

The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001394
(load all 54 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The genus of a permutation. The genus

$g(\pi)$ of a permutation

$\pi\in\mathfrak S_n$ is defined via the relation

$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$ where

$\zeta = (1,2,\dots,n)$ is the long cycle and

$z(\cdot)$ is the number of cycles in the permutation.

The following 785 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000201The number of leaf nodes in a binary tree. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000528The height of a poset. St000676The number of odd rises 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. 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. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000245The number of ascents of a permutation. St000336The leg major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000647The number of big descents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 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. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. 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. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001840The number of descents of a set partition. St001928The number of non-overlapping descents in a permutation. St000007The number of saliances of the permutation. St000025The number of initial rises 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. St000092The number of outer peaks of a permutation. St000105The number of blocks in the set partition. St000153The number of adjacent cycles of a permutation. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000251The number of nonsingleton blocks of a set partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000808The number of up steps of the associated bargraph. St000862The number of parts of the shifted shape of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001261The Castelnuovo-Mumford regularity of a graph. St001389The number of partitions of the same length below the given integer partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001581The achromatic number of a graph. St001597The Frobenius rank of a skew partition. St001732The number of peaks visible from the left. St001741The largest integer such that all patterns of this size are contained in the permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000004The major index of a permutation. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000094The depth of an ordered tree. St000141The maximum drop size of a permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000159The number of distinct parts of the integer partition. St000169The cocharge of a standard tableau. St000183The side length of the Durfee square of an integer partition. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000330The (standard) major index of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000358The number of occurrences of the pattern 31-2. 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. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000481The number of upper covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000546The number of global descents of a permutation. St000632The jump number of the poset. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St000670The reversal length 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001115The number of even descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero 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

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001214The aft of an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001280The number of parts of an integer partition that are at least two. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. 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. 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. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. 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. St000325The width of the tree associated to a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. 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. 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). St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St000099The number of valleys of a permutation, including the boundary. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000568The hook number of a binary tree. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St000023The number of inner peaks of a permutation. St000080The rank of the poset. 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000646The number of big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. 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. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001427The number of descents of a signed permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001812The biclique partition number of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000084The number of subtrees. 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. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000258The burning number of a graph. St000288The number of ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. 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. St000392The length of the longest run of ones in a binary word. 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. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000738The first entry in the last row of a standard tableau. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St000886The number of permutations with the same antidiagonal sums. St000925The number of topologically connected components of a set partition. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001062The maximal size of a block of a set partition. St001081The number of minimal length factorizations of a permutation into star transpositions. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 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. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001246The maximal difference between two consecutive entries of a permutation. 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. St001372The length of a longest cyclic run of ones of a binary word. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001642The Prague dimension 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. St001734The lettericity of a graph. St001814The number of partitions interlacing the given partition. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000039The number of crossings of a permutation. St000055The inversion sum of a permutation. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000150The floored half-sum of the multiplicities of a partition. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000224The sorting index of a permutation. St000228The size of a partition. St000242The number of indices that are not cyclical small weak excedances. St000254The nesting number of a set partition. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. 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. St000389The number of runs of ones of odd length in a binary word. St000459The hook length of the base cell of a partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000640The rank of the largest boolean interval in a poset. St000658The number of rises of length 2 of a Dyck path. St000663The number of right floats of a permutation. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000778The metric dimension of a graph. St000779The tier of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000919The number of maximal left branches of a binary tree. St000940The number of characters of the symmetric group whose value on the partition is zero. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. 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 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. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001358The largest degree of a regular subgraph of a graph. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001479The number of bridges of a graph. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. 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. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 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. St001962The proper pathwidth 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

$c_{n-1}$ . St001859The number of factors of the Stanley symmetric function associated with a permutation. St000216The absolute length of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000061The number of nodes on the left branch of a binary tree. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. 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. St000493The los statistic of a set partition. 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. St000794The mak of a permutation. St000873The aix statistic of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001388The number of non-attacking neighbors of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. 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. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001896The number of right descents of a signed permutations. St000486The number of cycles of length at least 3 of a permutation. St000648The number of 2-excedences of a permutation. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. 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. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St000317The cycle descent number of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001487The number of inner corners of a skew partition. St001589The nesting number of a perfect matching. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000217The number of occurrences of the pattern 312 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000732The number of double deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000761The number of ascents in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000630The length of the shortest palindromic decomposition of a binary word. 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

$eA$ . 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. 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

$g$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. 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. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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

$eA$ . 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000570The Edelman-Greene number of a permutation. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St000806The semiperimeter of the associated bargraph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000905The number of different multiplicities of parts of an integer composition. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . 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

$K[x]/(x^n)$ . St001220The width of a permutation. 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. 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. 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. 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. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001884The number of borders of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. 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. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000145The Dyson rank of a partition. St000225Difference between largest and smallest parts in a partition. St000252The number of nodes of degree 3 of a binary tree. St000379The number of Hamiltonian cycles in a graph. St000407The number of occurrences of the pattern 2143 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000516The number of stretching pairs of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000768The number of peaks in an integer composition. St000881The number of short braid edges in the graph of braid moves of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001584The area statistic between a Dyck path and its bounce path. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001856The number of edges in the reduced word graph of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000454The largest eigenvalue of a graph if it is integral. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000842The breadth of a permutation. St001624The breadth of a lattice. St000259The diameter of a connected graph. St001162The minimum jump of a permutation. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000651The maximal size of a rise in a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000260The radius of a connected graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. 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. St000402Half the size of the symmetry class of a permutation. St001545The second Elser number of a connected graph. St001621The number of atoms of a lattice. St000022The number of fixed points of a permutation. St000456The monochromatic index of a connected graph. St000891The number of distinct diagonal sums of a permutation matrix. St000396The register function (or Horton-Strahler number) of a binary tree. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001399The distinguishing number of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000731The number of double exceedences of a permutation. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001586The number of odd parts smaller than the largest even part in an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001870The number of positive entries followed by a negative entry in a signed permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St001569The maximal modular displacement of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001645The pebbling number of a connected graph. St000264The girth of a graph, which is not a tree. St000366The number of double descents of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St001118The acyclic chromatic index of a graph. 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

$K[x]/(x^n)$ . St001735The number of permutations with the same set of runs. St000563The number of overlapping pairs of blocks of a set partition. St000694The number of affine bounded permutations that project to a given permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000916The packing number of a graph. St001060The distinguishing index of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. 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. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000154The sum of the descent bottoms of a permutation. St000247The number of singleton blocks of a set partition. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000504The cardinality of the first block of a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000618The number of self-evacuating tableaux of given shape. St000654The first descent of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000989The number of final rises 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

$K[x]/(x^n)$ . St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001353The number of prime nodes in the modular decomposition of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001603The number of colourings of a polygon such that the multiplicities of a colour 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001810The number of fixed points of a permutation smaller than its largest moved point. St001871The number of triconnected components of a graph. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. 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. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001722The number of minimal chains with small intervals between a binary word and the top element. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000477The weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. 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. 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. 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. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St001568The smallest positive integer that does not appear twice in the 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. 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. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000502The number of successions of a set partitions. St000988The orbit size of a permutation under Foata's bijection. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001557The number of inversions of the second entry of a permutation. St001937The size of the center of a parking function. St000222The number of alignments in the permutation. St000232The number of crossings of a set partition. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000338The number of pixed points of a permutation. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000534The number of 2-rises of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000824The sum of the number of descents and the number of recoils of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001377The major index minus the number of inversions of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001535The number of cyclic alignments of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001705The number of occurrences of the pattern 2413 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001822The number of alignments of a signed permutation. St001841The number of inversions of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001911A descent variant minus the number of inversions. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St001811The Castelnuovo-Mumford regularity of a permutation. St000464The Schultz index of a connected graph. St000736The last entry in the first row of a semistandard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000942The number of critical left to right maxima of the parking functions. 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001889The size of the connectivity set of a signed permutation. St001948The number of augmented double ascents of a permutation. St000075The orbit size of a standard tableau under promotion. 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. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. 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. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001637The number of (upper) dissectors of a poset. St001684The reduced word complexity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001933The largest multiplicity of a part in an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000699The toughness times the least common multiple of 1,. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition.