- FindStat

Your data matches 32 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000445
(load all 34 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000475

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000248
(load all 10 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of anti-singletons of a set partition. An anti-singleton of a set partition

$S$ is an index

$i$ such that

$i$ and

$i+1$ (considered cyclically) are both in the same block of

$S$ . For noncrossing set partitions, this is also the number of singletons of the image of

$S$ under the Kreweras complement.

Matching statistic: St000674
(load all 8 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.

Matching statistic: St001126
(load all 8 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 89%

Values

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

Description

Number of simple module that are 1-regular in the corresponding Nakayama algebra.

Matching statistic: St000247
(load all 14 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 89%

Values

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

Description

The number of singleton blocks of a set partition.

Matching statistic: St000986
(load all 3 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 89%

Values

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

Description

The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.

Matching statistic: St000932
(load all 4 compositions to match this statistic)

Mapping

Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 89%

Values

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

Description

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

Matching statistic: St001484

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000215
(load all 13 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form

$(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic,

$\operatorname{adj_0}$ , counts adjacencies in the word with a zero appended.

$(\operatorname{adj_0}, \operatorname{des})$ and

$(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].

The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000022The number of fixed points of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000502The number of successions of a set partitions. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000894The trace of an alternating sign matrix. St000441The number of successions of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000237The number of small exceedances. St000214The number of adjacencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St000239The number of small weak excedances. St001061The number of indices that are both descents and recoils of a permutation. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000731The number of double exceedences of a permutation. St001948The number of augmented double ascents of a permutation.