- FindStat

Your data matches 135 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

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

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

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

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000013
(load all 5 compositions to match this statistic)

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000378

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => []
 => []
 => 0
[[2],[]]
 => []
 => []
 => 0
[[1,1],[]]
 => []
 => []
 => 0
[[2,1],[1]]
 => [1]
 => [1]
 => 1
[[3],[]]
 => []
 => []
 => 0
[[2,1],[]]
 => []
 => []
 => 0
[[3,1],[1]]
 => [1]
 => [1]
 => 1
[[2,2],[1]]
 => [1]
 => [1]
 => 1
[[3,2],[2]]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[]]
 => []
 => []
 => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[[2,1,1],[1]]
 => [1]
 => [1]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [3]
 => 2
[[4],[]]
 => []
 => []
 => 0
[[3,1],[]]
 => []
 => []
 => 0
[[4,1],[1]]
 => [1]
 => [1]
 => 1
[[2,2],[]]
 => []
 => []
 => 0
[[3,2],[1]]
 => [1]
 => [1]
 => 1
[[4,2],[2]]
 => [2]
 => [1,1]
 => 1
[[2,1,1],[]]
 => []
 => []
 => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[[3,1,1],[1]]
 => [1]
 => [1]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [3]
 => 2
[[3,3],[2]]
 => [2]
 => [1,1]
 => 1
[[4,3],[3]]
 => [3]
 => [1,1,1]
 => 1
[[2,2,1],[1]]
 => [1]
 => [1]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [3]
 => 2
[[3,2,1],[2]]
 => [2]
 => [1,1]
 => 1
[[4,3,1],[3,1]]
 => [3,1]
 => [2,1,1]
 => 2
[[2,2,2],[1,1]]
 => [1,1]
 => [2]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [4]
 => 2
[[3,2,2],[2,1]]
 => [2,1]
 => [3]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [5]
 => 2
[[1,1,1,1],[]]
 => []
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [2,1]
 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [2,2,1]
 => 3
[[2,1,1,1],[1]]
 => [1]
 => [1]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [2,2]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [3]
 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [5,1]
 => 3
[[5],[]]
 => []
 => []
 => 0
[[4,1],[]]
 => []
 => []
 => 0
[[5,1],[1]]
 => [1]
 => [1]
 => 1
[[3,2],[]]
 => []
 => []
 => 0
[[4,2],[1]]
 => [1]
 => [1]
 => 1
[[5,2],[2]]
 => [2]
 => [1,1]
 => 1
[[3,1,1],[]]
 => []
 => []
 => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [2]
 => 2
[[4,1,1],[1]]
 => [1]
 => [1]
 => 1
[[6,5,4,3,1],[5,4,3,1]]
 => [5,4,3,1]
 => [9,2,1,1]
 => ? ∊ {4,5}
[[5,4,4,3,2,1],[4,3,3,2,1]]
 => [4,3,3,2,1]
 => [7,2,2,2]
 => ? ∊ {4,5}

Description

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

$c$ in the diagram of an integer partition

$\lambda$ for which

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

Matching statistic: St000676
(load all 6 compositions to match this statistic)

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St000025

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000007

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => []
 => []
 => [] => 0
[[2],[]]
 => []
 => []
 => [] => 0
[[1,1],[]]
 => []
 => []
 => [] => 0
[[2,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[3],[]]
 => []
 => []
 => [] => 0
[[2,1],[]]
 => []
 => []
 => [] => 0
[[3,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[2,2],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[3,2],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[1,1,1],[]]
 => []
 => []
 => [] => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[2,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[4],[]]
 => []
 => []
 => [] => 0
[[3,1],[]]
 => []
 => []
 => [] => 0
[[4,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[2,2],[]]
 => []
 => []
 => [] => 0
[[3,2],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[4,2],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[2,1,1],[]]
 => []
 => []
 => [] => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[3,3],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[4,3],[3]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[[2,2,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[3,2,1],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[4,3,1],[3,1]]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[[2,2,2],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[[3,2,2],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 2
[[1,1,1,1],[]]
 => []
 => []
 => [] => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 3
[[2,1,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => 3
[[5],[]]
 => []
 => []
 => [] => 0
[[4,1],[]]
 => []
 => []
 => [] => 0
[[5,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[3,2],[]]
 => []
 => []
 => [] => 0
[[4,2],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[5,2],[2]]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[[3,1,1],[]]
 => []
 => []
 => [] => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[[4,1,1],[1]]
 => [1]
 => [[1]]
 => [1] => 1
[[6,5,3,2,1],[5,3,2,1]]
 => [5,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11]]
 => [11,9,10,6,7,8,1,2,3,4,5] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,2],[5,4,2]]
 => [5,4,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11]]
 => [10,11,6,7,8,9,1,2,3,4,5] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,2,1],[4,4,2,1]]
 => [4,4,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11]]
 => [11,9,10,5,6,7,8,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,2,1],[5,4,2,1]]
 => [5,4,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
 => [12,10,11,6,7,8,9,1,2,3,4,5] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3],[4,4,3]]
 => [4,4,3]
 => [[1,2,3,4],[5,6,7,8],[9,10,11]]
 => [9,10,11,5,6,7,8,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3],[5,4,3]]
 => [5,4,3]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
 => [10,11,12,6,7,8,9,1,2,3,4,5] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,1],[4,4,3,1]]
 => [4,4,3,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
 => [12,9,10,11,5,6,7,8,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,1],[4,3,3,1]]
 => [4,3,3,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11]]
 => [11,8,9,10,5,6,7,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,1],[5,4,3,1]]
 => [5,4,3,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
 => [13,10,11,12,6,7,8,9,1,2,3,4,5] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,4,3,2],[3,3,3,2]]
 => [3,3,3,2]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11]]
 => [10,11,7,8,9,4,5,6,1,2,3] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,2],[4,4,3,2]]
 => [4,4,3,2]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
 => [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,2],[4,3,3,2]]
 => [4,3,3,2]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
 => [11,12,8,9,10,5,6,7,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,3,2],[4,3,2,2]]
 => [4,3,2,2]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11]]
 => [10,11,8,9,5,6,7,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,2],[5,4,3,2]]
 => [5,4,3,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
 => [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,4,3,2,1],[3,3,3,2,1]]
 => [3,3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
 => [12,10,11,7,8,9,4,5,6,1,2,3] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,3,3,2,1],[3,3,2,2,1]]
 => [3,3,2,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
 => [11,9,10,7,8,4,5,6,1,2,3] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,2,1],[4,4,3,2,1]]
 => [4,4,3,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
 => [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,2,1],[4,3,3,2,1]]
 => [4,3,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
 => [13,11,12,8,9,10,5,6,7,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,3,2,1],[4,3,2,2,1]]
 => [4,3,2,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
 => [12,10,11,8,9,5,6,7,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,2,2,1],[4,3,2,1,1]]
 => [4,3,2,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
 => [11,10,8,9,5,6,7,1,2,3,4] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,2,1],[5,4,3,2,1]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000507
(load all 2 compositions to match this statistic)

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => []
 => []
 => []
 => 0
[[2],[]]
 => []
 => []
 => []
 => 0
[[1,1],[]]
 => []
 => []
 => []
 => 0
[[2,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3],[]]
 => []
 => []
 => []
 => 0
[[2,1],[]]
 => []
 => []
 => []
 => 0
[[3,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[2,2],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[1,1,1],[]]
 => []
 => []
 => []
 => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[2,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[4],[]]
 => []
 => []
 => []
 => 0
[[3,1],[]]
 => []
 => []
 => []
 => 0
[[4,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[2,2],[]]
 => []
 => []
 => []
 => 0
[[3,2],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[4,2],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[2,1,1],[]]
 => []
 => []
 => []
 => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[3,3],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[4,3],[3]]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1
[[2,2,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[3,2,1],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[4,3,1],[3,1]]
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[[2,2,2],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[[3,2,2],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
[[1,1,1,1],[]]
 => []
 => []
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 3
[[2,1,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 3
[[5],[]]
 => []
 => []
 => []
 => 0
[[4,1],[]]
 => []
 => []
 => []
 => 0
[[5,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2],[]]
 => []
 => []
 => []
 => 0
[[4,2],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[5,2],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[[3,1,1],[]]
 => []
 => []
 => []
 => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[[4,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[6,5,3,2,1],[5,3,2,1]]
 => [5,3,2,1]
 => [[1,3,6,10,11],[2,5,9],[4,8],[7]]
 => [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,2],[5,4,2]]
 => [5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,2,1],[4,4,2,1]]
 => [4,4,2,1]
 => [[1,3,6,7],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,2,1],[5,4,2,1]]
 => [5,4,2,1]
 => [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3],[4,4,3]]
 => [4,4,3]
 => [[1,2,3,7],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3],[5,4,3]]
 => [5,4,3]
 => [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,1],[4,4,3,1]]
 => [4,4,3,1]
 => [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
 => [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,1],[4,3,3,1]]
 => [4,3,3,1]
 => [[1,3,4,11],[2,6,7],[5,9,10],[8]]
 => [[1,2,5,8],[3,6,9],[4,7,10],[11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,1],[5,4,3,1]]
 => [5,4,3,1]
 => [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
 => [[1,2,5,9],[3,6,10],[4,7,11],[8,12],[13]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,4,3,2],[3,3,3,2]]
 => [3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,2],[4,4,3,2]]
 => [4,4,3,2]
 => [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
 => [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,2],[4,3,3,2]]
 => [4,3,3,2]
 => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,3,2],[4,3,2,2]]
 => [4,3,2,2]
 => [[1,2,7,11],[3,4,10],[5,6],[8,9]]
 => [[1,3,5,8],[2,4,6,9],[7,10],[11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,2],[5,4,3,2]]
 => [5,4,3,2]
 => [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
 => [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13],[14]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,4,3,2,1],[3,3,3,2,1]]
 => [3,3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,3,3,2,1],[3,3,2,2,1]]
 => [3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,2,1],[4,4,3,2,1]]
 => [4,4,3,2,1]
 => [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,2,1],[4,3,3,2,1]]
 => [4,3,3,2,1]
 => [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,3,2,1],[4,3,2,2,1]]
 => [4,3,2,2,1]
 => [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,2,2,1],[4,3,2,1,1]]
 => [4,3,2,1,1]
 => [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
 => [[1,2,3,5,8],[4,6,9],[7,10],[11]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,2,1],[5,4,3,2,1]]
 => [5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000745

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[[1],[]]
 => []
 => []
 => []
 => 0
[[2],[]]
 => []
 => []
 => []
 => 0
[[1,1],[]]
 => []
 => []
 => []
 => 0
[[2,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3],[]]
 => []
 => []
 => []
 => 0
[[2,1],[]]
 => []
 => []
 => []
 => 0
[[3,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[2,2],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[[1,1,1],[]]
 => []
 => []
 => []
 => 0
[[2,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[[2,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[4],[]]
 => []
 => []
 => []
 => 0
[[3,1],[]]
 => []
 => []
 => []
 => 0
[[4,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[2,2],[]]
 => []
 => []
 => []
 => 0
[[3,2],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[4,2],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[[2,1,1],[]]
 => []
 => []
 => []
 => 0
[[3,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[[3,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[3,3],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[[4,3],[3]]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[[2,2,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[3,2,1],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
[[2,2,2],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[[3,3,2],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[[3,2,2],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
[[1,1,1,1],[]]
 => []
 => []
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 2
[[2,1,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
[[3,2,1,1],[2,1]]
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[1,4,6],[2,5],[3]]
 => 3
[[5],[]]
 => []
 => []
 => []
 => 0
[[4,1],[]]
 => []
 => []
 => []
 => 0
[[5,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[3,2],[]]
 => []
 => []
 => []
 => 0
[[4,2],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[5,2],[2]]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[[3,1,1],[]]
 => []
 => []
 => []
 => 0
[[4,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[[4,1,1],[1]]
 => [1]
 => [[1]]
 => [[1]]
 => 1
[[6,5,3,2,1],[5,3,2,1]]
 => [5,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11]]
 => [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,2],[5,4,2]]
 => [5,4,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11]]
 => [[1,6,10],[2,7,11],[3,8],[4,9],[5]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,2,1],[4,4,2,1]]
 => [4,4,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11]]
 => [[1,5,9,11],[2,6,10],[3,7],[4,8]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,2,1],[5,4,2,1]]
 => [5,4,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
 => [[1,6,10,12],[2,7,11],[3,8],[4,9],[5]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3],[4,4,3]]
 => [4,4,3]
 => [[1,2,3,4],[5,6,7,8],[9,10,11]]
 => [[1,5,9],[2,6,10],[3,7,11],[4,8]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3],[5,4,3]]
 => [5,4,3]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
 => [[1,6,10],[2,7,11],[3,8,12],[4,9],[5]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,1],[4,4,3,1]]
 => [4,4,3,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
 => [[1,5,9,12],[2,6,10],[3,7,11],[4,8]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,1],[4,3,3,1]]
 => [4,3,3,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11]]
 => [[1,5,8,11],[2,6,9],[3,7,10],[4]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,1],[5,4,3,1]]
 => [5,4,3,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
 => [[1,6,10,13],[2,7,11],[3,8,12],[4,9],[5]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,4,3,2],[3,3,3,2]]
 => [3,3,3,2]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,2],[4,4,3,2]]
 => [4,4,3,2]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
 => [[1,5,9,12],[2,6,10,13],[3,7,11],[4,8]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,2],[4,3,3,2]]
 => [4,3,3,2]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
 => [[1,5,8,11],[2,6,9,12],[3,7,10],[4]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,3,2],[4,3,2,2]]
 => [4,3,2,2]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11]]
 => [[1,5,8,10],[2,6,9,11],[3,7],[4]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,2],[5,4,3,2]]
 => [5,4,3,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
 => [[1,6,10,13],[2,7,11,14],[3,8,12],[4,9],[5]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,4,3,2,1],[3,3,3,2,1]]
 => [3,3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
 => [[1,4,7,10,12],[2,5,8,11],[3,6,9]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[4,4,3,3,2,1],[3,3,2,2,1]]
 => [3,3,2,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
 => [[1,4,7,9,11],[2,5,8,10],[3,6]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,5,4,3,2,1],[4,4,3,2,1]]
 => [4,4,3,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
 => [[1,5,9,12,14],[2,6,10,13],[3,7,11],[4,8]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,4,3,2,1],[4,3,3,2,1]]
 => [4,3,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
 => [[1,5,8,11,13],[2,6,9,12],[3,7,10],[4]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,3,2,1],[4,3,2,2,1]]
 => [4,3,2,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
 => [[1,5,8,10,12],[2,6,9,11],[3,7],[4]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[5,4,3,2,2,1],[4,3,2,1,1]]
 => [4,3,2,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
 => [[1,5,8,10,11],[2,6,9],[3,7],[4]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}
[[6,5,4,3,2,1],[5,4,3,2,1]]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,6,10,13,15],[2,7,11,14],[3,8,12],[4,9],[5]]
 => ? ∊ {3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5}

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St000288
(load all 8 compositions to match this statistic)

Mapping

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 93%●distinct values known / distinct values provided: 83%

Values

[[1],[]]
 => []
 =>  => ? = 0
[[2],[]]
 => []
 =>  => ? ∊ {0,0}
[[1,1],[]]
 => []
 =>  => ? ∊ {0,0}
[[2,1],[1]]
 => [1]
 => 10 => 1
[[3],[]]
 => []
 =>  => ? ∊ {0,0,0}
[[2,1],[]]
 => []
 =>  => ? ∊ {0,0,0}
[[3,1],[1]]
 => [1]
 => 10 => 1
[[2,2],[1]]
 => [1]
 => 10 => 1
[[3,2],[2]]
 => [2]
 => 100 => 1
[[1,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0}
[[2,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[[2,1,1],[1]]
 => [1]
 => 10 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => 1010 => 2
[[4],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0}
[[3,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0}
[[4,1],[1]]
 => [1]
 => 10 => 1
[[2,2],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0}
[[3,2],[1]]
 => [1]
 => 10 => 1
[[4,2],[2]]
 => [2]
 => 100 => 1
[[2,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0}
[[3,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[[3,1,1],[1]]
 => [1]
 => 10 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => 1010 => 2
[[3,3],[2]]
 => [2]
 => 100 => 1
[[4,3],[3]]
 => [3]
 => 1000 => 1
[[2,2,1],[1]]
 => [1]
 => 10 => 1
[[3,3,1],[2,1]]
 => [2,1]
 => 1010 => 2
[[3,2,1],[2]]
 => [2]
 => 100 => 1
[[4,3,1],[3,1]]
 => [3,1]
 => 10010 => 2
[[2,2,2],[1,1]]
 => [1,1]
 => 110 => 2
[[3,3,2],[2,2]]
 => [2,2]
 => 1100 => 2
[[3,2,2],[2,1]]
 => [2,1]
 => 1010 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => 10100 => 2
[[1,1,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0}
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 3
[[2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 11010 => 3
[[2,1,1,1],[1]]
 => [1]
 => 10 => 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 10110 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => 1010 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 101010 => 3
[[5],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0}
[[4,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0}
[[5,1],[1]]
 => [1]
 => 10 => 1
[[3,2],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0}
[[4,2],[1]]
 => [1]
 => 10 => 1
[[5,2],[2]]
 => [2]
 => 100 => 1
[[3,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0}
[[4,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[[4,1,1],[1]]
 => [1]
 => 10 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => 1010 => 2
[[3,3],[1]]
 => [1]
 => 10 => 1
[[4,3],[2]]
 => [2]
 => 100 => 1
[[5,3],[3]]
 => [3]
 => 1000 => 1
[[2,2,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0}
[[3,3,1],[1,1]]
 => [1,1]
 => 110 => 2
[[3,2,1],[1]]
 => [1]
 => 10 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => 1010 => 2
[[4,2,1],[2]]
 => [2]
 => 100 => 1
[[5,3,1],[3,1]]
 => [3,1]
 => 10010 => 2
[[3,2,2],[1,1]]
 => [1,1]
 => 110 => 2
[[4,3,2],[2,2]]
 => [2,2]
 => 1100 => 2
[[4,2,2],[2,1]]
 => [2,1]
 => 1010 => 2
[[5,3,2],[3,2]]
 => [3,2]
 => 10100 => 2
[[2,1,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0}
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 3
[[3,2,1,1],[1,1]]
 => [1,1]
 => 110 => 2
[[1,1,1,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0}
[[6],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[5,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[4,2],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[4,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[3,3],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[3,2,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[3,1,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[2,2,2],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[2,2,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[2,1,1,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}
[[1,1,1,1,1,1],[]]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0}

Description

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

The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000141The maximum drop size of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000691The number of changes of a binary word. St000031The number of cycles in the cycle decomposition of a permutation. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001809The index of the step at the first peak of maximal height in a Dyck path. St000382The first part of an integer composition. St000740The last entry of a permutation. St000470The number of runs in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000381The largest part of an integer composition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000443The number of long tunnels of a Dyck path. 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. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000325The width of the tree associated to a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000991The number of right-to-left minima of a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000840The number of closers smaller than the largest opener in a perfect matching. St000925The number of topologically connected components of a set partition. St000702The number of weak deficiencies of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000738The first entry in the last row of a standard tableau. St001462The number of factors of a standard tableaux under concatenation. St000489The number of cycles of a permutation of length at most 3. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000144The pyramid weight of the Dyck path. St000225Difference between largest and smallest parts in a partition. St000006The dinv of a Dyck path. St000383The last part of an integer composition. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000746The number of pairs with odd minimum in a perfect matching. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000105The number of blocks in the set partition. St001589The nesting number of a perfect matching. St001637The number of (upper) dissectors of a poset. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. 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

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001435The number of missing boxes in the first row. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001875The number of simple modules with projective dimension at most 1. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001645The pebbling number of a connected graph. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001668The number of points of the poset minus the width of the poset. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St001839The number of excedances of a set partition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000782The indicator function of whether a given perfect matching is an L & P matching. St001778The largest greatest common divisor of an element and its image in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.