- FindStat

Your data matches 123 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000980

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0,1,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [3]
 => [1,0,1,0,1,0]
 => 0

Description

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path

$111011010000$ has three peaks in positions

$03, 15, 26$ . The boxes below

$03$ are

$01,02,\textbf{12}$ , the boxes below

$15$ are

$\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$ , and the boxes below

$26$ are

$\textbf{23},\textbf{24},25,\textbf{34},35,45$ . We thus obtain the four boxes in positions

$12,23,24,34$ that are below at least two peaks.

Matching statistic: St001435
(load all 7 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of missing boxes in the first row.

Matching statistic: St001438
(load all 7 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of missing boxes of a skew partition.

Matching statistic: St001487
(load all 7 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of inner corners of a skew partition.

Matching statistic: St001490
(load all 161 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of connected components of a skew partition.

Matching statistic: St001811

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

[1,0,1,0,1,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 0
[1,0,1,1,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,1,0,0,1,0]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ? = 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ? = 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ? = 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ? = 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ? = 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ? = 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ? = 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ? = 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ? = 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
 => [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
 => [11,9,10,7,8,4,5,6,1,2,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,4,5,1,2,3] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
 => [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
 => [11,10,7,8,9,4,5,6,1,2,3] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
 => [11,10,8,9,6,7,1,2,3,4,5] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
 => [11,10,9,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 0
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [10,11,8,9,6,7,1,2,3,4,5] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
 => [12,9,10,11,6,7,8,1,2,3,4,5] => ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [10,7,8,9,4,5,6,1,2,3] => ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ? = 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ? = 0
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10]]
 => [10,9,6,7,8,1,2,3,4,5] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 0
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 0
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 0

Description

The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation

$\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety''

$X_\sigma$ . Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for

$\sigma$ . It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].

Matching statistic: St000181
(load all 140 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

[1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[5,3,2,2,1],[]]
 => ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[3,3,2,2,1],[]]
 => ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [[3,2,2,2,1],[]]
 => ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [[2,2,2,2,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[5,3,3,1,1],[]]
 => ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11),(10,12),(11,12)],13)
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[3,3,3,1,1],[]]
 => ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[5,2,2,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[3,2,2,1,1],[]]
 => ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
 => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[5,3,1,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[5,1,1,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [[5,3,2,2],[]]
 => ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2]
 => [[3,3,2,2],[]]
 => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[5,2,2,2],[]]
 => ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2]
 => [[3,2,2,2],[]]
 => ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [[5,3,3,1],[]]
 => ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10),(9,11),(10,11)],12)
 => ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1]
 => [[3,3,3,1],[]]
 => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [[5,2,2,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 2 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [[5,3,1,1],[]]
 => ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1

Description

The number of connected components of the Hasse diagram for the poset.

Matching statistic: St001208
(load all 5 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

[1,0,1,0,1,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
 => [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
 => [11,9,10,7,8,4,5,6,1,2,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,4,5,1,2,3] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
 => [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
 => [11,10,7,8,9,4,5,6,1,2,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
 => [11,10,8,9,6,7,1,2,3,4,5] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
 => [11,10,9,6,7,8,1,2,3,4,5] => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [10,11,8,9,6,7,1,2,3,4,5] => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
 => [12,9,10,11,6,7,8,1,2,3,4,5] => ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [10,7,8,9,4,5,6,1,2,3] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => ? = 2 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ? = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10]]
 => [10,9,6,7,8,1,2,3,4,5] => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1 = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1 = 0 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1 = 0 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1 = 0 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1 = 0 + 1

Description

The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .

Matching statistic: St001890
(load all 140 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 21%●distinct values known / distinct values provided: 11%

Values

[1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[5,3,2,2,1],[]]
 => ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[3,3,2,2,1],[]]
 => ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [[3,2,2,2,1],[]]
 => ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [[2,2,2,2,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
 => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[5,3,3,1,1],[]]
 => ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11),(10,12),(11,12)],13)
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[3,3,3,1,1],[]]
 => ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[5,2,2,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[3,2,2,1,1],[]]
 => ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
 => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[5,3,1,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [[5,1,1,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [[5,3,2,2],[]]
 => ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2]
 => [[3,3,2,2],[]]
 => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[5,2,2,2],[]]
 => ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2]
 => [[3,2,2,2],[]]
 => ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [[5,3,3,1],[]]
 => ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10),(9,11),(10,11)],12)
 => ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1]
 => [[3,3,3,1],[]]
 => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [[5,2,2,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 2 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ? = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [[5,3,1,1],[]]
 => ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[2],[]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1 = 0 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1

Description

The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value

$\mu(x, y)$ is equal to the signed sum of chains from

$x$ to

$y$ , where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.

Matching statistic: St001371
(load all 2 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 18%●distinct values known / distinct values provided: 11%

Values

[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => 10111010010100 => ? = 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 111010010100 => ? = 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 10101110010100 => ? = 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => ? = 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 10101011010100 => ? = 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? = 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? = 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? = 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => ? = 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? = 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? = 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 101011101101000100 => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => 11101101000100 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => 10111101000100 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 111101000100 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 101011111000010100 => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 11111000010100 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => 101010111100010100 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => 10111100010100 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 111100010100 => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 101011101001010100 => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 11101001010100 => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => 10111001010100 => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 111001010100 => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 101010101101010100 => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 1010101101010100 => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 10101101010100 => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 101101010100 => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? = 0
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => 1010111011010000 => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 111011010000 => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => 1010101111010000 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => 1010111110000100 => ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 111110000100 => ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => 1010101111000100 => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => 1010 => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

The following 113 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001730The number of times the path corresponding to a binary word crosses the base line. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St000297The number of leading ones in a binary word. St000052The number of valleys of a Dyck path not on the x-axis. St001434The number of negative sum pairs of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001568The smallest positive integer that does not appear twice in the partition. St000068The number of minimal elements in a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000878The number of ones minus the number of zeros of a binary word. St001429The number of negative entries in a signed permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St001947The number of ties in a parking function. St000741The Colin de Verdière graph invariant. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001555The order of a signed permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000054The first entry of the permutation. St001260The permanent of an alternating sign matrix. St000078The number of alternating sign matrices whose left key is the permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. 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

$eA$ . St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000296The length of the symmetric border of a binary word. St000546The number of global descents of a permutation. St000629The defect of a binary word. St000787The number of flips required to make a perfect matching noncrossing. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000383The last part of an integer composition. St000627The exponent of a binary word. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000381The largest part of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000842The breadth of a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000753The Grundy value for the game of Kayles on a binary word. St000234The number of global ascents of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St001381The fertility of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000862The number of parts of the shifted shape of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000298The order dimension or Dushnik-Miller dimension of a poset. St000983The length of the longest alternating subword. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000648The number of 2-excedences of a permutation. St000731The number of double exceedences of a permutation. St001130The number of two successive successions in a permutation. St000717The number of ordinal summands of a poset. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001394The genus of a permutation. St000237The number of small exceedances. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000124The cardinality of the preimage of the Simion-Schmidt map. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000374The number of exclusive right-to-left minima of a permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000007The number of saliances of the permutation. St000451The length of the longest pattern of the form k 1 2.