- FindStat

Your data matches 150 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001446

Mapping

St001446: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of rows in the plane partition.

Matching statistic: St001447

Mapping

St001447: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Height of the base box of a plane partition.

Matching statistic: St001460

Mapping

St001460: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of columns of a plane partition.

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000157
(load all 2 compositions to match this statistic)

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000329
(load all 2 compositions to match this statistic)

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

Matching statistic: St000147
(load all 2 compositions to match this statistic)

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000288
(load all 10 compositions to match this statistic)

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1]
 => 10 => 1 = 0 + 1
[[1],[1]]
 => [1,1]
 => 110 => 2 = 1 + 1
[[2]]
 => [2]
 => 100 => 1 = 0 + 1
[[1,1]]
 => [2]
 => 100 => 1 = 0 + 1
[[1],[1],[1]]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[[2],[1]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[1,1],[1]]
 => [2,1]
 => 1010 => 2 = 1 + 1
[[3]]
 => [3]
 => 1000 => 1 = 0 + 1
[[2,1]]
 => [3]
 => 1000 => 1 = 0 + 1
[[1,1,1]]
 => [3]
 => 1000 => 1 = 0 + 1
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => 11110 => 4 = 3 + 1
[[2],[1],[1]]
 => [2,1,1]
 => 10110 => 3 = 2 + 1
[[2],[2]]
 => [2,2]
 => 1100 => 2 = 1 + 1
[[1,1],[1],[1]]
 => [2,1,1]
 => 10110 => 3 = 2 + 1
[[1,1],[1,1]]
 => [2,2]
 => 1100 => 2 = 1 + 1
[[3],[1]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[2,1],[1]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[1,1,1],[1]]
 => [3,1]
 => 10010 => 2 = 1 + 1
[[4]]
 => [4]
 => 10000 => 1 = 0 + 1
[[3,1]]
 => [4]
 => 10000 => 1 = 0 + 1
[[2,2]]
 => [4]
 => 10000 => 1 = 0 + 1
[[2,1,1]]
 => [4]
 => 10000 => 1 = 0 + 1
[[1,1,1,1]]
 => [4]
 => 10000 => 1 = 0 + 1
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => 111110 => 5 = 4 + 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => 101110 => 4 = 3 + 1
[[2],[2],[1]]
 => [2,2,1]
 => 11010 => 3 = 2 + 1
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => 101110 => 4 = 3 + 1
[[1,1],[1,1],[1]]
 => [2,2,1]
 => 11010 => 3 = 2 + 1
[[3],[1],[1]]
 => [3,1,1]
 => 100110 => 3 = 2 + 1
[[3],[2]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[2,1],[1],[1]]
 => [3,1,1]
 => 100110 => 3 = 2 + 1
[[2,1],[2]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[2,1],[1,1]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[1,1,1],[1],[1]]
 => [3,1,1]
 => 100110 => 3 = 2 + 1
[[1,1,1],[1,1]]
 => [3,2]
 => 10100 => 2 = 1 + 1
[[4],[1]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[3,1],[1]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[2,2],[1]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[2,1,1],[1]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[1,1,1,1],[1]]
 => [4,1]
 => 100010 => 2 = 1 + 1
[[5]]
 => [5]
 => 100000 => 1 = 0 + 1
[[4,1]]
 => [5]
 => 100000 => 1 = 0 + 1
[[3,2]]
 => [5]
 => 100000 => 1 = 0 + 1
[[3,1,1]]
 => [5]
 => 100000 => 1 = 0 + 1
[[2,2,1]]
 => [5]
 => 100000 => 1 = 0 + 1
[[2,1,1,1]]
 => [5]
 => 100000 => 1 = 0 + 1
[[1,1,1,1,1]]
 => [5]
 => 100000 => 1 = 0 + 1
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => 1111110 => 6 = 5 + 1
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => 1011110 => 5 = 4 + 1
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => 110110 => 4 = 3 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000378

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$ . See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].

Matching statistic: St000733
(load all 3 compositions to match this statistic)

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The row containing the largest entry of a standard tableau.

The following 140 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St000546The number of global descents of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001777The number of weak descents in an integer composition. St000007The number of saliances of the permutation. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001530The depth of a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000216The absolute length of a permutation. St000354The number of recoils 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. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000710The number of big deficiencies of a permutation. St000921The number of internal inversions of a binary word. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001152The number of pairs with even minimum in a perfect matching. 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. St001485The modular major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000681The Grundy value of Chomp on Ferrers diagrams. St000993The multiplicity of the largest part of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000744The length of the path to the largest entry in a standard Young tableau. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000984The number of boxes below precisely one peak. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition.