- FindStat

Your data matches 38 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000504
(load all 3 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of the first block of a set partition. The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].

Matching statistic: St000382

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

Matching statistic: St000932
(load all 64 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1

Description

The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

Matching statistic: St000502
(load all 8 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => {{1},{2}}
 => 0 = 1 - 1
[1,1,0,0]
 => {{1,2}}
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3},{4,5,6,7,8}}
 => ? = 6 - 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3},{4,5,6,7,8}}
 => ? = 7 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 7 - 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => {{1,5,6,7,8},{2,3,4}}
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 7 - 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => {{1,6,7,8},{2,3,4,5}}
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => {{1,2,6,7,8},{3,4,5}}
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => {{1,7,8},{2,3,4,5,6}}
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => {{1,2,7,8},{3,4,5,6}}
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => {{1,2,8},{3,4,5,6,7}}
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => {{1,2,3,8},{4,5,6,7}}
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => {{1,2,3,4,8},{5,6,7}}
 => ? = 6 - 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: St001067
(load all 64 compositions to match this statistic)

Mapping

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 95%●distinct values known / distinct values provided: 88%

Values

[1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 7 - 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1

Description

The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.

Matching statistic: St001189
(load all 13 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 95%●distinct values known / distinct values provided: 88%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 6 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7 - 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 7 - 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 7 - 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7 - 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => ? = 6 - 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 7 - 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 6 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8 - 1

Description

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001484
(load all 4 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,0]
 => []
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1]
 => ? = 6 - 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1]
 => ? = 4 - 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2]
 => ? = 4 - 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,1]
 => ? = 5 - 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1]
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1,1]
 => ? = 3 - 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2]
 => ? = 4 - 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1,1]
 => ? = 3 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => ? = 5 - 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1]
 => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,1,1]
 => ? = 3 - 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => ? = 3 - 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2]
 => ? = 4 - 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => ? = 3 - 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,1]
 => ? = 5 - 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => ? = 5 - 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => ? = 5 - 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1]
 => ? = 7 - 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1]
 => ? = 7 - 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,1,1]
 => ? = 6 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1]
 => ? = 7 - 1
[1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1,1]
 => ? = 6 - 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,2,1]
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1]
 => ? = 7 - 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1,1]
 => ? = 6 - 1
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,2,1]
 => ? = 6 - 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,3,2,1]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1]
 => ? = 7 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1,1]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,2,1]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,3,2,1]
 => ? = 6 - 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,4,3,2,1]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,1]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2,1]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2,1]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2,1]
 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2,1]
 => ? = 6 - 1

Description

The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.

Matching statistic: St000445
(load all 7 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 3 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
 => ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 4 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 4 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4 - 1

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St001223
(load all 67 compositions to match this statistic)

Mapping

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 75%

Values

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

Description

Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.

Matching statistic: St001640
(load all 11 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 88%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [2,1] => 0 = 1 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,3,1] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [3,4,5,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,4,5,1,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [3,4,1,5,2] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [4,5,1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,3,5,1,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,5,1,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,4,1,5,3] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,1,5,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,5,1,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [3,1,5,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [3,4,1,2,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [2,4,1,3,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [3,1,4,2,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,3,1,5,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,4,5,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,3,5,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,2,5,3] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,1,7,2,3,4,5] => [2,4,5,6,7,1,3] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,1,2,7,3,4,5] => [2,3,5,6,7,1,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,1,7,2,3,4] => [3,5,6,7,1,2,4] => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [5,1,7,2,3,4,6] => [2,4,5,6,1,7,3] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,1,6,7,2,3,4] => [2,5,6,7,1,3,4] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,6,1,7,2,3,5] => [3,5,6,1,7,2,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => [5,6,7,1,2,3,4] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,1,2,3,7,4,5] => [2,3,4,6,7,1,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,1,2,7,3,4] => [3,4,6,7,1,2,5] => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,1,6,2,7,3,4] => [2,4,6,7,1,3,5] => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [4,6,1,2,7,3,5] => [3,4,6,1,7,2,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => [4,6,7,1,2,3,5] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [5,1,2,7,3,4,6] => [2,3,5,6,1,7,4] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [4,5,1,7,2,3,6] => [3,5,6,1,2,7,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [4,1,7,2,3,5,6] => [2,4,5,1,6,7,3] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,7,1,2,4,5,6] => [3,4,1,5,6,7,2] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,6,7,1,2,4,5] => [4,5,1,6,7,2,3] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [4,1,6,7,2,3,5] => [2,5,6,1,7,3,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [3,6,1,7,2,4,5] => [3,5,1,6,7,2,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [3,5,7,1,2,4,6] => [4,5,1,6,2,7,3] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,5,6,7,1,2,4] => [5,6,1,7,2,3,4] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,1,2,6,7,3,4] => [2,3,6,7,1,4,5] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,1,6,7,2,3] => [3,6,7,1,2,4,5] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [4,1,6,2,7,3,5] => [2,4,6,1,7,3,5] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [3,6,1,2,7,4,5] => [3,4,1,6,7,2,5] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [3,5,6,1,7,2,4] => [4,6,1,7,2,3,5] => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [4,1,5,7,2,3,6] => [2,5,6,1,3,7,4] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [3,5,1,7,2,4,6] => [3,5,1,6,2,7,4] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [3,4,7,1,2,5,6] => [4,5,1,2,6,7,3] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,1,5,6,7,2,3] => [2,6,7,1,3,4,5] => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [3,5,1,6,7,2,4] => [3,6,1,7,2,4,5] => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [3,4,6,1,7,2,5] => [4,6,1,2,7,3,5] => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,7,1,2,6] => [5,6,1,2,3,7,4] => ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,1,2,3,4,7,5] => [2,3,4,5,7,1,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,1,2,3,7,4] => [3,4,5,7,1,2,6] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [5,1,6,2,3,7,4] => [2,4,5,7,1,3,6] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [4,6,1,2,3,7,5] => [3,4,5,1,7,2,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,1,2,7,3] => [4,5,7,1,2,3,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [5,1,2,6,3,7,4] => [2,3,5,7,1,4,6] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [4,5,1,6,2,7,3] => [3,5,7,1,2,4,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [4,1,6,2,3,7,5] => [2,4,5,1,7,3,6] => ? = 2 - 1

Description

The number of ascent tops in the permutation such that all smaller elements appear before.

The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001061The number of indices that are both descents and recoils of a permutation. St000441The number of successions of a permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St000931The number of occurrences of the pattern UUU in a Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000215The number of adjacencies of a permutation, zero appended. St000731The number of double exceedences of a permutation. St000365The number of double ascents of a permutation. St001948The number of augmented double ascents of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St000317The cycle descent number of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St001530The depth of a Dyck path.