- FindStat

Your data matches 102 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001913

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of preimages of an integer partition in Bulgarian solitaire. A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row. Partitions without preimages are called garden of eden partitions [1].

Matching statistic: St001568
(load all 5 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 33%

Values

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

Description

The smallest positive integer that does not appear twice in the partition.

Matching statistic: St000478
(load all 3 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => 0 = 1 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 1 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => 0 = 1 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => 0 = 1 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 0 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => 0 = 1 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => 0 = 1 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => ? = 2 - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 0 = 1 - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [4,3,3]
 => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => 0 = 1 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1]
 => ? = 1 - 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1]
 => ? = 1 - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,1]
 => ? = 1 - 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2]
 => ? = 1 - 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1]
 => 0 = 1 - 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1]
 => ? = 0 - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,1]
 => ? = 1 - 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1]
 => ? = 1 - 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [4,3,3,1]
 => 0 = 1 - 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2]
 => ? = 2 - 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2]
 => ? = 1 - 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2]
 => 0 = 1 - 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1]
 => ? = 1 - 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [5,4,4,1]
 => ? = 1 - 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => [5,5,3,1]
 => ? = 1 - 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => ? = 0 - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,1]
 => ? = 0 - 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1]
 => ? = 0 - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [4,4,2,1]
 => 0 = 1 - 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1]
 => ? = 1 - 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2]
 => ? = 1 - 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,3,1]
 => 0 = 1 - 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2]
 => ? = 1 - 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1]
 => ? = 2 - 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2]
 => ? = 2 - 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2]
 => 0 = 1 - 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2]
 => ? = 2 - 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2]
 => ? = 1 - 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2]
 => ? = 1 - 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3]
 => ? = 1 - 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2]
 => ? = 1 - 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [4,4,4,1]
 => ? = 1 - 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2]
 => ? = 2 - 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3]
 => ? = 1 - 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3]
 => ? = 1 - 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [4,3,3]
 => 0 = 1 - 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [4,4,2]
 => 0 = 1 - 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1]
 => ? = 1 - 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1]
 => ? = 1 - 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,3,1]
 => ? = 1 - 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,2]
 => ? = 1 - 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1]
 => ? = 0 - 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,2,1]
 => ? = 1 - 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,3,1]
 => ? = 1 - 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,2]
 => ? = 2 - 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,2,2]
 => ? = 1 - 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
 => [6,4,3,2,2]
 => ? = 1 - 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
 => [6,5,5,2,1]
 => ? = 1 - 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
 => [6,6,4,2,1]
 => ? = 1 - 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1]
 => ? = 0 - 1
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,5,3,2,1]
 => ? = 0 - 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,4,2,1]
 => ? = 0 - 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,3,1]
 => ? = 1 - 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,5,3,2,2]
 => ? = 1 - 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,3,1]
 => ? = 1 - 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[7,1,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1

Description

Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1]

$\sum_{\pi\in GG_1(r)} w_1(\pi)$ equals the number of partitions of

$r$ whose odd parts are all distinct.

$GG_1(r)$ is the set of partitions of

$r$ where consecutive entries differ by at least

$2$ , and consecutive even entries differ by at least

$4$ .

Matching statistic: St000510
(load all 7 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => 0 = 1 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 1 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => 0 = 1 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => 0 = 1 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 0 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => 0 = 1 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => 0 = 1 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => ? = 2 - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 0 = 1 - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [4,3,3]
 => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => 0 = 1 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1]
 => ? = 1 - 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1]
 => ? = 1 - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,1]
 => ? = 1 - 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2]
 => ? = 1 - 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1]
 => 0 = 1 - 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1]
 => ? = 0 - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,1]
 => ? = 1 - 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1]
 => ? = 1 - 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [4,3,3,1]
 => 0 = 1 - 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2]
 => ? = 2 - 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2]
 => ? = 1 - 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2]
 => 0 = 1 - 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1]
 => ? = 1 - 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [5,4,4,1]
 => ? = 1 - 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => [5,5,3,1]
 => ? = 1 - 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => ? = 0 - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,1]
 => ? = 0 - 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1]
 => ? = 0 - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [4,4,2,1]
 => 0 = 1 - 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1]
 => ? = 1 - 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2]
 => ? = 1 - 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,3,1]
 => 0 = 1 - 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2]
 => ? = 1 - 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1]
 => ? = 2 - 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2]
 => ? = 2 - 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2]
 => 0 = 1 - 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2]
 => ? = 2 - 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2]
 => ? = 1 - 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2]
 => ? = 1 - 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3]
 => ? = 1 - 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2]
 => ? = 1 - 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [4,4,4,1]
 => ? = 1 - 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2]
 => ? = 2 - 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3]
 => ? = 1 - 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3]
 => ? = 1 - 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [4,3,3]
 => 0 = 1 - 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [4,4,2]
 => 0 = 1 - 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1]
 => ? = 1 - 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1]
 => ? = 1 - 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,3,1]
 => ? = 1 - 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,2]
 => ? = 1 - 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1]
 => ? = 0 - 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,2,1]
 => ? = 1 - 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,3,1]
 => ? = 1 - 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,2]
 => ? = 2 - 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,2,2]
 => ? = 1 - 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
 => [6,4,3,2,2]
 => ? = 1 - 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
 => [6,5,5,2,1]
 => ? = 1 - 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
 => [6,6,4,2,1]
 => ? = 1 - 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1]
 => ? = 0 - 1
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,5,3,2,1]
 => ? = 0 - 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,4,2,1]
 => ? = 0 - 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,3,1]
 => ? = 1 - 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,5,3,2,2]
 => ? = 1 - 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,3,1]
 => ? = 1 - 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[7,1,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1

Description

The number of invariant oriented cycles when acting with a permutation of given cycle type.

Matching statistic: St000929
(load all 7 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => 0 = 1 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 1 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => 0 = 1 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => 0 = 1 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 0 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => 0 = 1 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => 0 = 1 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => ? = 2 - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 0 = 1 - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [4,3,3]
 => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => 0 = 1 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1]
 => ? = 1 - 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1]
 => ? = 1 - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,1]
 => ? = 1 - 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2]
 => ? = 1 - 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1]
 => 0 = 1 - 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1]
 => ? = 0 - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,1]
 => ? = 1 - 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1]
 => ? = 1 - 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [4,3,3,1]
 => 0 = 1 - 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2]
 => ? = 2 - 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2]
 => ? = 1 - 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2]
 => 0 = 1 - 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1]
 => ? = 1 - 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [5,4,4,1]
 => ? = 1 - 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => [5,5,3,1]
 => ? = 1 - 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => 0 = 1 - 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => ? = 0 - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,1]
 => ? = 0 - 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1]
 => ? = 0 - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [4,4,2,1]
 => 0 = 1 - 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1]
 => ? = 1 - 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2]
 => ? = 1 - 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,3,1]
 => 0 = 1 - 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [3,3,1]
 => 0 = 1 - 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2]
 => ? = 1 - 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1]
 => ? = 2 - 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2]
 => ? = 2 - 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2]
 => 0 = 1 - 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2]
 => ? = 2 - 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2]
 => ? = 1 - 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [4,2,2]
 => 0 = 1 - 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2]
 => ? = 1 - 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3]
 => ? = 1 - 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2]
 => ? = 1 - 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => [4,4,4,1]
 => ? = 1 - 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2]
 => ? = 2 - 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3]
 => ? = 1 - 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3]
 => ? = 1 - 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [4,3,3]
 => 0 = 1 - 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [4,4,2]
 => 0 = 1 - 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,1]
 => ? = 1 - 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,1]
 => ? = 1 - 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,3,1]
 => ? = 1 - 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,2]
 => ? = 1 - 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,1]
 => ? = 0 - 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,2,1]
 => ? = 1 - 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,3,1]
 => ? = 1 - 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,2]
 => ? = 2 - 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,2,2]
 => ? = 1 - 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
 => [6,4,3,2,2]
 => ? = 1 - 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
 => [6,5,5,2,1]
 => ? = 1 - 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
 => [6,6,4,2,1]
 => ? = 1 - 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1]
 => ? = 0 - 1
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,5,3,2,1]
 => ? = 0 - 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,4,2,1]
 => ? = 0 - 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,3,1]
 => ? = 1 - 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,5,3,2,2]
 => ? = 1 - 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,3,1]
 => ? = 1 - 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1
[7,1,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2]
 => 0 = 1 - 1
[8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2]
 => 0 = 1 - 1

Description

The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

Matching statistic: St000049
(load all 2 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000049: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? = 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => ? = 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [19]
 => ? = 0
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [20]
 => ? = 0
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 0
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 0
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 2
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6]
 => 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 2
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1

Description

The number of set partitions whose sorted block sizes correspond to the partition.

Matching statistic: St000146

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? = 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => ? = 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [19]
 => ? = 0
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [20]
 => ? = 0
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 0
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 0
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 2
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6]
 => 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 2
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1

Description

The Andrews-Garvan crank of a partition. If

$\pi$ is a partition, let

$l(\pi)$ be its length (number of parts),

$\omega(\pi)$ be the number of parts equal to 1, and

$\mu(\pi)$ be the number of parts larger than

$\omega(\pi)$ . The crank is then defined by

$c(\pi) = \begin{cases} l(\pi) &\text{if $\omega(\pi)=0$}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases}$ This statistic was defined in [1] to explain Ramanujan's partition congruence

$p(11n+6) \equiv 0 \pmod{11}$ in the same way as the Dyson rank ([[St000145]]) explains the congruences

$p(5n+4) \equiv 0 \pmod{5}$ and

$p(7n+5) \equiv 0 \pmod{7}.$

Matching statistic: St000256
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? = 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => ? = 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [19]
 => ? = 0
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [20]
 => ? = 0
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 0
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 0
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 2
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6]
 => 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 2
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1

Description

The number of parts from which one can substract 2 and still get an integer partition.

Matching statistic: St000275
(load all 2 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000275: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? = 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => ? = 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [19]
 => ? = 0
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [20]
 => ? = 0
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 0
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 0
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 2
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6]
 => 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 2
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1

Description

Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition.

Matching statistic: St000781
(load all 4 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%

Values

[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? = 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [11]
 => ? = 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [9]
 => 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [19]
 => ? = 0
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [20]
 => ? = 0
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 0
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 0
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [16]
 => ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [17]
 => ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 2
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6]
 => 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [18]
 => ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [12]
 => ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [13]
 => ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [14]
 => ? = 2
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => 1
[5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [11]
 => ? = 1
[1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
[1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
[1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1

Description

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

The following 92 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000783The side length of the largest staircase partition fitting into a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000143The largest repeated part of a partition. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001214The aft of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001890The maximum magnitude of the Möbius function of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001335The cardinality of a minimal cycle-isolating set of a graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000455The second largest eigenvalue of a graph if it is integral. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St001307The number of induced stars on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000315The number of isolated vertices of a graph. St000644The number of graphs with given frequency partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001578The minimal number of edges to add or remove to make a graph a line graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St000917The open packing number of a graph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000258The burning number of a graph. St000259The diameter of a connected graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000918The 2-limited packing number of a graph. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St001111The weak 2-dynamic chromatic number of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000264The girth of a graph, which is not a tree. St001674The number of vertices of the largest induced star graph in the graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001260The permanent of an alternating sign matrix. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2.