- FindStat

Your data matches 160 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000502

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of successions of a set partitions. This is the number of indices

$i$ such that

$i$ and

$i+1$ belonging to the same block.

Matching statistic: St001693

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001693: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The excess length of a longest path consisting of elements and blocks of a set partition. Let

$p$ be a set partition of

$\{1,\dots,n\}$ . Let

$G$ be the graph with edges

$(i,i+1)$ for

$i\in\{1,\dots,n-1\}$ and

$(i, b)$ , whenever

$i$ is an element of a non-singleton block

$b\in p$ . Then this statistic records the length of the longest path from

$1$ to

$n$ in

$G$ , reduced by

$n$ . Conjecturally, a longest path has more than

$n$ vertices provided that the set partition has no singletons.

Matching statistic: St001440
(load all 5 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 83%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.

Matching statistic: St001940
(load all 13 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 83%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of distinct parts that are equal to their multiplicity in the integer partition.

Matching statistic: St000940
(load all 6 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of characters of the symmetric group whose value on the partition is zero. The maximal value for any given size is recorded in [2].

Matching statistic: St001124
(load all 15 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 70%●distinct values known / distinct values provided: 50%

Values

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

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

Matching statistic: St000225
(load all 5 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%

Values

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

Description

Difference between largest and smallest parts in a partition.

Matching statistic: St000319
(load all 2 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%

Values

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

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320
(load all 2 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%

Values

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

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St000506
(load all 3 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry

$i$ such that

$i+1$ appears to the right or above

$i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

The following 150 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001280The number of parts of an integer partition that are at least two. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001525The number of symmetric hooks on the diagonal of a 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. St001939The number of parts that are equal to their multiplicity in the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000929The constant term of the character polynomial of an integer partition. St000120The number of left tunnels of a Dyck path. St000159The number of distinct parts of the integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001484The number of singletons of an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001730The number of times the path corresponding to a binary word crosses the base line. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). 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. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000661The number of rises of length 3 of a Dyck path. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St000214The number of adjacencies of a permutation. 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. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001498The normalised height of a Nakayama algebra with magnitude 1. St000137The Grundy value of an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001593This is the number of standard Young tableaux of the given shifted shape. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000237The number of small exceedances. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001877Number of indecomposable injective modules with projective dimension 2. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000650The number of 3-rises of a permutation. St000872The number of very big descents of a permutation. St000618The number of self-evacuating tableaux of given shape. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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. St001568The smallest positive integer that does not appear twice in the partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001403The number of vertical separators in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000624The normalized sum of the minimal distances to a greater element. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001570The minimal number of edges to add to make a graph Hamiltonian. St000241The number of cyclical small excedances. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001948The number of augmented double ascents of a permutation. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001820The size of the image of the pop stack sorting operator. St001095The number of non-isomorphic posets with precisely one further covering relation. St000366The number of double descents of a permutation. St001964The interval resolution global dimension of a poset. St000383The last part of an integer composition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000022The number of fixed points of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001846The number of elements which do not have a complement in the lattice. St000908The length of the shortest maximal antichain in a poset. St001330The hat guessing number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001810The number of fixed points of a permutation smaller than its largest moved point. St000741The Colin de Verdière graph invariant. St001050The number of terminal closers of a set partition.