- FindStat

Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001452

Mapping

St001452: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of even parts in the plane partition.

Matching statistic: St000142

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000142: 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]
 => [2]
 => 1
[[2]]
 => [2]
 => [1,1]
 => 0
[[1,1]]
 => [2]
 => [1,1]
 => 0
[[1],[1],[1]]
 => [1,1,1]
 => [3]
 => 0
[[2],[1]]
 => [2,1]
 => [2,1]
 => 1
[[1,1],[1]]
 => [2,1]
 => [2,1]
 => 1
[[3]]
 => [3]
 => [1,1,1]
 => 0
[[2,1]]
 => [3]
 => [1,1,1]
 => 0
[[1,1,1]]
 => [3]
 => [1,1,1]
 => 0
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [4]
 => 1
[[2],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 0
[[2],[2]]
 => [2,2]
 => [2,2]
 => 2
[[1,1],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 0
[[1,1],[1,1]]
 => [2,2]
 => [2,2]
 => 2
[[3],[1]]
 => [3,1]
 => [2,1,1]
 => 1
[[2,1],[1]]
 => [3,1]
 => [2,1,1]
 => 1
[[1,1,1],[1]]
 => [3,1]
 => [2,1,1]
 => 1
[[4]]
 => [4]
 => [1,1,1,1]
 => 0
[[3,1]]
 => [4]
 => [1,1,1,1]
 => 0
[[2,2]]
 => [4]
 => [1,1,1,1]
 => 0
[[2,1,1]]
 => [4]
 => [1,1,1,1]
 => 0
[[1,1,1,1]]
 => [4]
 => [1,1,1,1]
 => 0
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [5]
 => 0
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [3,2]
 => 1
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 1
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [3,2]
 => 1
[[3],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 0
[[3],[2]]
 => [3,2]
 => [2,2,1]
 => 2
[[2,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 0
[[2,1],[2]]
 => [3,2]
 => [2,2,1]
 => 2
[[2,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 2
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 0
[[1,1,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 2
[[4],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[3,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[2,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[2,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[1,1,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[5]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[4,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[3,2]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [6]
 => 1
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [5,1]
 => 0
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [4,2]
 => 2

Description

The number of even parts of a partition.

Matching statistic: St000992

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The alternating sum of the parts of an integer partition. For a partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ , this is

$\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$ .

Matching statistic: St000148

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd parts of a partition.

Matching statistic: St000149

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: 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]
 => [2]
 => [2]
 => 1
[[2]]
 => [2]
 => [1,1]
 => [1,1]
 => 0
[[1,1]]
 => [2]
 => [1,1]
 => [1,1]
 => 0
[[1],[1],[1]]
 => [1,1,1]
 => [3]
 => [2,1]
 => 0
[[2],[1]]
 => [2,1]
 => [2,1]
 => [3]
 => 1
[[1,1],[1]]
 => [2,1]
 => [2,1]
 => [3]
 => 1
[[3]]
 => [3]
 => [1,1,1]
 => [1,1,1]
 => 0
[[2,1]]
 => [3]
 => [1,1,1]
 => [1,1,1]
 => 0
[[1,1,1]]
 => [3]
 => [1,1,1]
 => [1,1,1]
 => 0
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[[2],[1],[1]]
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 0
[[2],[2]]
 => [2,2]
 => [2,2]
 => [4]
 => 2
[[1,1],[1],[1]]
 => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 0
[[1,1],[1,1]]
 => [2,2]
 => [2,2]
 => [4]
 => 2
[[3],[1]]
 => [3,1]
 => [2,1,1]
 => [3,1]
 => 1
[[2,1],[1]]
 => [3,1]
 => [2,1,1]
 => [3,1]
 => 1
[[1,1,1],[1]]
 => [3,1]
 => [2,1,1]
 => [3,1]
 => 1
[[4]]
 => [4]
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
[[3,1]]
 => [4]
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
[[2,2]]
 => [4]
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
[[2,1,1]]
 => [4]
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,1,1,1]]
 => [4]
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [5]
 => [2,2,1]
 => 0
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => [3,2]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [3,2]
 => [4,1]
 => 1
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => [3,2]
 => 1
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [3,2]
 => [4,1]
 => 1
[[3],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => 0
[[3],[2]]
 => [3,2]
 => [2,2,1]
 => [5]
 => 2
[[2,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => 0
[[2,1],[2]]
 => [3,2]
 => [2,2,1]
 => [5]
 => 2
[[2,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => [5]
 => 2
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => 0
[[1,1,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => [5]
 => 2
[[4],[1]]
 => [4,1]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[[3,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[[2,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[[2,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[[1,1,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[[5]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[[4,1]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[[3,2]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [6]
 => [2,2,2]
 => 1
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [5,1]
 => [2,2,1,1]
 => 0
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [4,2]
 => [4,2]
 => 2

Description

The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].

Matching statistic: St000150

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: 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]
 => [2]
 => [1,1]
 => 1
[[2]]
 => [2]
 => [1,1]
 => [2]
 => 0
[[1,1]]
 => [2]
 => [1,1]
 => [2]
 => 0
[[1],[1],[1]]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[[2],[1]]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[[1,1],[1]]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[[3]]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[[2,1]]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[[1,1,1]]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[[2],[1],[1]]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[[2],[2]]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 2
[[1,1],[1],[1]]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[[1,1],[1,1]]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 2
[[3],[1]]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[[2,1],[1]]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[[1,1,1],[1]]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[[4]]
 => [4]
 => [1,1,1,1]
 => [4]
 => 0
[[3,1]]
 => [4]
 => [1,1,1,1]
 => [4]
 => 0
[[2,2]]
 => [4]
 => [1,1,1,1]
 => [4]
 => 0
[[2,1,1]]
 => [4]
 => [1,1,1,1]
 => [4]
 => 0
[[1,1,1,1]]
 => [4]
 => [1,1,1,1]
 => [4]
 => 0
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [5]
 => [5]
 => 0
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => [2,2,1]
 => 1
[[2],[2],[1]]
 => [2,2,1]
 => [3,2]
 => [3,1,1]
 => 1
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => [2,2,1]
 => 1
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [3,2]
 => [3,1,1]
 => 1
[[3],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 0
[[3],[2]]
 => [3,2]
 => [2,2,1]
 => [1,1,1,1,1]
 => 2
[[2,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 0
[[2,1],[2]]
 => [3,2]
 => [2,2,1]
 => [1,1,1,1,1]
 => 2
[[2,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => [1,1,1,1,1]
 => 2
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 0
[[1,1,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => [1,1,1,1,1]
 => 2
[[4],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[3,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[2,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[2,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[1,1,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => 1
[[5]]
 => [5]
 => [1,1,1,1,1]
 => [4,1]
 => 0
[[4,1]]
 => [5]
 => [1,1,1,1,1]
 => [4,1]
 => 0
[[3,2]]
 => [5]
 => [1,1,1,1,1]
 => [4,1]
 => 0
[[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [4,1]
 => 0
[[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => [4,1]
 => 0
[[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [4,1]
 => 0
[[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => [4,1]
 => 0
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [6]
 => [3,3]
 => 1
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [5,1]
 => [5,1]
 => 0
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => 2

Description

The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].

Matching statistic: St000566

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if

$\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is

$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$

Matching statistic: St000668

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%

Values

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

Description

The least common multiple of the parts of the partition.

Matching statistic: St000698

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer

$k$ , one associates a

$k$ -core to a partition by repeatedly removing all rim hooks of size

$k$ . This statistic counts the

$2$ -rim hooks that are removed in this process to obtain a

$2$ -core.

Matching statistic: St000714

Mapping

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of

$GL_2$ .

The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.