- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000932

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 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: St001067

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001223

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1,0,1,0]
 => 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: St001484

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1]
 => 1
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,1]
 => 0
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,1]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [3]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [2,2]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,1]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,1,1]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,1]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [2,1]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,1]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,1]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,1]
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [3]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [2,1]
 => 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1]
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [2]
 => 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [2,2]
 => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,1,1]
 => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,1]
 => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [2,1]
 => 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1]
 => 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: St001086

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern

$132$ , where the matched entries are all adjacent.

Matching statistic: St000541

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation

$\pi$ of length

$n$ , this is the number of indices

$2 \leq j \leq n$ such that for all

$1 \leq i < j$ , the pair

$(i,j)$ is an inversion of

$\pi$ .

Matching statistic: St000542

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of left-to-right-minima of a permutation. An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a left-to-right-minimum if there does not exist a j < i such that

$\sigma_j < \sigma_i$ .

Matching statistic: St000991

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].