- FindStat

Your data matches 389 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000183
(load all 2 compositions to match this statistic)

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The side length of the Durfee square of an integer partition. Given a partition

$\lambda = (\lambda_1,\ldots,\lambda_n)$ , the Durfee square is the largest partition

$(s^s)$ whose diagram fits inside the diagram of

$\lambda$ . In symbols,

$s = \max\{ i \mid \lambda_i \geq i \}$ . This is also known as the Frobenius rank.

Matching statistic: St001175
(load all 2 compositions to match this statistic)

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St000024

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000147

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000208

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given

$\lambda$ count how many ''integer partitions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].

Matching statistic: St000321

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

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

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

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

$k$ .

Matching statistic: St000345

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of refinements of a partition. A partition

$\lambda$ refines a partition

$\mu$ if the parts of

$\mu$ can be subdivided to obtain the parts of

$\lambda$ .

Matching statistic: St000442

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000755

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence

$f(n)=\sum_{p\in\lambda} f(n-p).$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition

$(2,1)$ corresponds to the recurrence

$f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial

$x^2-x-1$ , which has two real roots.

Matching statistic: St000913

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ways to refine the partition into singletons. For example there is only one way to refine

$[2,2]$ :

$[2,2] > [2,1,1] > [1,1,1,1]$ . However, there are two ways to refine

$[3,2]$ :

$[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and

$[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$ . In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition. The sequence of values on the partitions with only one part is [[A002846]].

The following 379 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001280The number of parts of an integer partition that are at least two. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001389The number of partitions of the same length below the given integer partition. St001597The Frobenius rank of a skew partition. St000013The height of a Dyck path. St000017The number of inversions of a standard tableau. St000057The Shynar inversion number of a standard tableau. St000142The number of even parts of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even parts of a partition. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001587Half of the largest even part of an integer partition. St001596The number of two-by-two squares inside a skew partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000390The number of runs of ones in a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001196The global dimension of

$A$ minus the global dimension of

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

$eA$ . St000649The number of 3-excedences of a permutation. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001471The magnitude of a Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001256Number of simple reflexive modules that are 2-stable reflexive. St001556The number of inversions of the third entry of a permutation. St000764The number of strong records in an integer composition. St000657The smallest part of an integer composition. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000383The last part of an integer composition. St000655The length of the minimal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000700The protection number of an ordered tree. St000701The protection number of a binary tree. St000703The number of deficiencies of a permutation. St000990The first ascent of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001075The minimal size of a block of a set partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000439The position of the first down step of a Dyck path. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

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

$c_0$ to

$c_{n-1}$ . St000989The number of final rises of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001310The number of induced diamond graphs in a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001513The number of nested exceedences of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000617The number of global maxima of a Dyck path. St000210Minimum over maximum difference of elements in cycles. St000352The Elizalde-Pak rank 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. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

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

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000546The number of global descents of a permutation. St000883The number of longest increasing subsequences of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000035The number of left outer peaks of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000654The first descent of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000221The number of strong fixed points of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001578The minimal number of edges to add or remove to make a graph a line graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000756The sum of the positions of the left to right maxima of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St001260The permanent of an alternating sign matrix. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000090The variation of a composition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St000508Eigenvalues of the random-to-random operator acting on a simple module. St001371The length of the longest Yamanouchi prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000069The number of maximal elements of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001948The number of augmented double ascents of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001344The neighbouring number of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001430The number of positive entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001488The number of corners of a skew partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001868The number of alignments of type NE of a signed permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000068The number of minimal elements in a poset. St001555The order of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000850The number of 1/2-balanced pairs in a poset. St000911The number of maximal antichains of maximal size in a poset. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001884The number of borders of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000295The length of the border of a binary word. St000298The order dimension or Dushnik-Miller dimension of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001399The distinguishing number of a poset. St001423The number of distinct cubes in a binary word. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001851The number of Hecke atoms of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000717The number of ordinal summands of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001892The flag excedance statistic of a signed permutation. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000264The girth of a graph, which is not a tree. St001964The interval resolution global dimension of a poset. St001618The cardinality of the Frattini sublattice of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001410The minimal entry of a semistandard tableau. St001510The number of self-evacuating linear extensions of a finite poset. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001268The size of the largest ordinal summand in the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000635The number of strictly order preserving maps of a poset into itself. St000307The number of rowmotion orbits of a poset. St001621The number of atoms of a lattice. St001625The Möbius invariant of a lattice. St000741The Colin de Verdière graph invariant. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. St000084The number of subtrees. St000100The number of linear extensions of a poset. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001623The number of doubly irreducible elements of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001754The number of tolerances of a finite lattice. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000189The number of elements in the poset. St000327The number of cover relations in a poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001645The pebbling number of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000180The number of chains of a poset. St000400The path length of an ordered tree. St001909The number of interval-closed sets of a poset. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000529The number of permutations whose descent word is the given binary word. St000416The number of inequivalent increasing trees of an ordered tree. St000634The number of endomorphisms of a poset. St000410The tree factorial of an ordered tree. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.