- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000148
(load all 2 compositions to match this statistic)

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd parts of a partition.

Matching statistic: St000288

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2,1] => [[2,2],[1]]
 => [1]
 => 1 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 11 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => 1 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => 0 => 0
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0 => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 11 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 01 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => 1 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 00 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => 0 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => 1 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0 => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 01 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 1 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 00 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1 => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1111 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 011 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 11 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 001 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 01 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 11 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => 1 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 000 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 00 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 10 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => 0 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 11 => 2
[4,2] => [[5,4],[3]]
 => [3]
 => 1 => 1
[5,1] => [[5,5],[4]]
 => [4]
 => 0 => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => 0 => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => 01 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 1 => 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St001372

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2,1] => [[2,2],[1]]
 => [1]
 => 1 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 11 => 2
[2,2] => [[3,2],[1]]
 => [1]
 => 1 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => 0 => 0
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0 => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 11 => 2
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 01 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => 1 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 00 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => 0 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => 1 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0 => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 01 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 1 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 00 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1 => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1111 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 011 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 11 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 001 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 01 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 11 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => 1 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 000 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 00 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 10 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => 0 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 11 => 2
[4,2] => [[5,4],[3]]
 => [3]
 => 1 => 1
[5,1] => [[5,5],[4]]
 => [4]
 => 0 => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 1 => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => 0 => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => 111 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 11 => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => 01 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 1 => 1

Description

The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.

Matching statistic: St000142

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of even parts of a partition.

Matching statistic: St000992

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

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

Description

The alternating sum of the parts of an integer partition. For a partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ , this is

$\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$ .

Matching statistic: St001172

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of 1-rises at odd height of a Dyck path.