- FindStat

Your data matches 136 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001465
(load all 4 compositions to match this statistic)

Mapping

St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacent transpositions in the cycle decomposition of a permutation.

Matching statistic: St000214

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacencies of a permutation. An adjacency of a permutation

$\pi$ is an index

$i$ such that

$\pi(i)-1 = \pi(i+1)$ . Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].

Matching statistic: St000237

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St001714
(load all 3 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. In particular, partitions with statistic

$0$ are wide partitions.

Matching statistic: St000205
(load all 18 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%

Values

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

Description

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

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

$w$ (weight) there are, such that

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

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St000206
(load all 16 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%

Values

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

Description

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

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

$w$ (weight) there are, such that

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

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

Matching statistic: St001175
(load all 19 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%

Values

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

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: St000621
(load all 6 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition

$\lambda \vdash n$ by the number of standard tableaux

$T$ of shape

$\lambda$ such that

$\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].

Matching statistic: St001912

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001912: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 84%●distinct values known / distinct values provided: 50%

Values

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

Description

The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.

Matching statistic: St001498
(load all 17 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 82%●distinct values known / distinct values provided: 25%

Values

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

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

The following 126 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000017The number of inversions of a standard tableau. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. 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. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix 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. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001139The number of occurrences of hills of size 2 in a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St000941The number of characters of the symmetric group whose value on the partition is even. St001651The Frankl number of a lattice. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000699The toughness times the least common multiple of 1,. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000781The number of proper colouring schemes of a Ferrers diagram. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000315The number of isolated vertices of a graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000454The largest eigenvalue of a graph if it is integral. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000665The number of rafts of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001060The distinguishing index of a graph.