- FindStat

Your data matches 180 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000395
(load all 66 compositions to match this statistic)

Mapping

St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001018
(load all 62 compositions to match this statistic)

Mapping

St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001218
(load all 45 compositions to match this statistic)

Mapping

St001218: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 3 = 1 + 2
[1,0,1,0]
 => 4 = 2 + 2
[1,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,0]
 => 5 = 3 + 2
[1,0,1,1,0,0]
 => 5 = 3 + 2
[1,1,0,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,0,0]
 => 6 = 4 + 2
[1,1,1,0,0,0]
 => 5 = 3 + 2
[1,0,1,0,1,0,1,0]
 => 6 = 4 + 2
[1,0,1,0,1,1,0,0]
 => 6 = 4 + 2
[1,0,1,1,0,0,1,0]
 => 6 = 4 + 2
[1,0,1,1,1,0,0,0]
 => 6 = 4 + 2
[1,1,0,0,1,0,1,0]
 => 6 = 4 + 2
[1,1,0,0,1,1,0,0]
 => 6 = 4 + 2
[1,1,1,0,0,0,1,0]
 => 6 = 4 + 2
[1,1,1,1,0,0,0,0]
 => 6 = 4 + 2
[1,0,1,0,1,0,1,0,1,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,1,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,1,0,0,1,0]
 => 7 = 5 + 2
[1,0,1,0,1,1,1,0,0,0]
 => 7 = 5 + 2
[1,0,1,1,0,0,1,0,1,0]
 => 7 = 5 + 2
[1,0,1,1,0,0,1,1,0,0]
 => 7 = 5 + 2
[1,0,1,1,1,0,0,0,1,0]
 => 7 = 5 + 2
[1,0,1,1,1,1,0,0,0,0]
 => 7 = 5 + 2
[1,1,0,0,1,0,1,0,1,0]
 => 7 = 5 + 2
[1,1,0,0,1,0,1,1,0,0]
 => 7 = 5 + 2
[1,1,0,0,1,1,0,0,1,0]
 => 7 = 5 + 2
[1,1,0,0,1,1,1,0,0,0]
 => 7 = 5 + 2
[1,1,1,0,0,0,1,0,1,0]
 => 7 = 5 + 2
[1,1,1,0,0,0,1,1,0,0]
 => 7 = 5 + 2
[1,1,1,1,0,0,0,0,1,0]
 => 7 = 5 + 2
[1,1,1,1,1,0,0,0,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 8 = 6 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 8 = 6 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 8 = 6 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 8 = 6 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => 8 = 6 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => 8 = 6 + 2

Description

Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. It returns zero in case there is no such k.

Matching statistic: St000229
(load all 11 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. This is, for a set partition

$P = \{B_1,\ldots,B_k\}$ of

$\{1,\ldots,n\}$ , the statistic is

$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$ This statistic is called ''dimension index'' in [2]

Matching statistic: St000394
(load all 13 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: 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]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 5
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 5
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 5
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 5
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[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]
 => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => 6
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => 6
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => 6
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => 6
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => 6

Description

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

Matching statistic: St001034
(load all 34 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].

Matching statistic: St000189
(load all 3 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000189: Posets ⟶ 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]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6

Description

The number of elements in the poset.

Matching statistic: St000476
(load all 7 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley

$v$ in a Dyck path

$D$ there is a corresponding tunnel, which is the factor

$T_v = s_i\dots s_j$ of

$D$ where

$s_i$ is the step after the first intersection of

$D$ with the line

$y = ht(v)$ to the left of

$s_j$ . This statistic is

$\sum_v (j_v-i_v)/2.$

Matching statistic: St000728
(load all 9 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dimension of a set partition. This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks. A slightly shifted definition of the dimension is [[St000572]].

Matching statistic: St001348
(load all 35 compositions to match this statistic)

Mapping

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The bounce of the parallelogram polyomino associated with the Dyck path. A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].

The following 170 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001717The largest size of an interval in a poset. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000031The number of cycles in the cycle decomposition of a permutation. St000081The number of edges of a graph. St000087The number of induced subgraphs. St000228The size of a partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000479The Ramsey number of a graph. St000636The hull number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000867The sum of the hook lengths in the first row of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001441The number of non-empty connected induced subgraphs of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000108The number of partitions contained in the given partition. St000151The number of facets in the chain polytope of the poset. St000238The number of indices that are not small weak excedances. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000438The position of the last up step in a Dyck path. St000446The disorder of a permutation. St000532The total number of rook placements on a Ferrers board. St000868The aid statistic in the sense of Shareshian-Wachs. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001664The number of non-isomorphic subposets of a poset. St001759The Rajchgot index of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000656The number of cuts of a poset. St000520The number of patterns in a permutation. St000770The major index of an integer partition when read from bottom to top. St000681The Grundy value of Chomp on Ferrers diagrams. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000794The mak of a permutation. St000795The mad of a permutation. St001875The number of simple modules with projective dimension at most 1. St001651The Frankl number of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001645The pebbling number of a connected graph. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000144The pyramid weight of the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000912The number of maximal antichains in a poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000744The length of the path to the largest entry in a standard Young tableau. St001497The position of the largest weak excedence of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000246The number of non-inversions of a permutation. St000883The number of longest increasing subsequences of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St001397Number of pairs of incomparable elements in a finite poset. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000030The sum of the descent differences of a permutations. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St001869The maximum cut size of a graph. St000067The inversion number of the alternating sign matrix. St000224The sorting index of a permutation. St000332The positive inversions of an alternating sign matrix. St000384The maximal part of the shifted composition of an integer partition. St001428The number of B-inversions of a signed permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000653The last descent of a permutation. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001280The number of parts of an integer partition that are at least two. St000058The order of a permutation. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000680The Grundy value for Hackendot on posets. St001894The depth of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001861The number of Bruhat lower covers of a permutation. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St001727The number of invisible inversions of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000906The length of the shortest maximal chain in a poset. St000474Dyson's crank of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000624The normalized sum of the minimal distances to a greater element. St000080The rank of the poset. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000702The number of weak deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001439The number of even weak deficiencies and of odd weak exceedences. St001416The length of a longest palindromic factor of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000806The semiperimeter of the associated bargraph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000632The jump number of the poset. St001820The size of the image of the pop stack sorting operator. St000527The width of the poset. St000528The height of a poset. St000845The maximal number of elements covered by an element in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000779The tier of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001812The biclique partition number of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001555The order of a signed permutation. St001782The order of rowmotion on the set of order ideals of a poset.