Your data matches 23 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001280
(load all 3 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [2]
 => 1
[1,1,0,0]
 => [2,1] => [1,1]
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [3]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,1,1]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [4,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,2]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [4,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [3,2]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000147
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => 1
[1,1,0,0]
 => [2,1] => [2]
 => []
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3]
 => []
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => []
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => [2]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 1
Description
The largest part of an integer partition.
Matching statistic: St000251
(load all 4 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000251: 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}}
 => 1
[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}}
 => 1
[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] => [[1,2],[3]]
 => {{1,2},{3}}
 => 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}}
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1
[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}}
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[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] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 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}}
 => 1
[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}}
 => 1
[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}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1
[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}}
 => 1
[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}}
 => 1
[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}}
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{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}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[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}}
 => 1
[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}}
 => 2
[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}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => 2
[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}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
[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}}
 => 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000319
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 99%●distinct values known / distinct values provided: 75%
Values
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => [2]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2]
 => [2]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 0 - 1
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 99%●distinct values known / distinct values provided: 75%
Values
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => [2]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2]
 => [2]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 0 - 1
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001918
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 99%●distinct values known / distinct values provided: 75%
Values
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => [2]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,1]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2]
 => [2]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 0 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 0 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.
Matching statistic: St000668
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 94%●distinct values known / distinct values provided: 75%
Values
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => ? = 1
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? = 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => ? = 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => ? = 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => ? = 1
[1,1,1,0,0,0]
 => [3,2,1] => [3]
 => []
 => ? = 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => ? = 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => ? = 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => ? = 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => [1]
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => [2]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,1]
 => [1]
 => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2]
 => [2]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [3,2]
 => [2]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => [4,1]
 => [1]
 => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [3,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [5,1]
 => [1]
 => ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => [5,1]
 => [1]
 => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [6,2,5,4,3,1] => [5,1]
 => [1]
 => ? = 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,5,4,2,1] => [5,1]
 => [1]
 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [5,1]
 => [1]
 => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => [5,1]
 => [1]
 => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [6,4,3,2,5,1] => [5,1]
 => [1]
 => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [6,4,3,5,2,1] => [5,1]
 => [1]
 => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,5,3,4,2,1] => [5,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [6,1]
 => [1]
 => ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,6,5,4,3,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [7,2,6,5,4,3,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [7,3,6,5,4,2,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [7,6,3,5,4,2,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [7,5,4,3,2,6,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [7,5,4,3,6,2,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [7,6,4,3,5,2,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,6,4,5,3,2,1] => [6,1]
 => [1]
 => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 0
Description
The least common multiple of the parts of the partition.
Matching statistic: St000845
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [2,1] => [2,1] => ([],2)
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => ([(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => ([],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
 => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,6,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,5,4] => [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [4,7,1,2,3,5,6] => ([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [4,6,1,2,3,5,7] => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [4,6,7,1,2,3,5] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [7,4,6,1,2,3,5] => ([(1,4),(2,3),(2,6),(4,5),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,6,3] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,5,7,3] => [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
 => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,5,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,6,5,3] => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [5,7,4,6,1,2,3] => ([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,4,7,3] => [5,6,4,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
 => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,4,3] => [5,6,7,4,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,6,4,3] => [7,5,6,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,5,4,3] => [6,7,5,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [3,7,1,2,4,5,6] => ([(0,4),(1,3),(1,6),(4,6),(5,2),(6,5)],7)
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [3,6,1,2,4,5,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [3,6,7,1,2,4,5] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [7,3,6,1,2,4,5] => ([(1,4),(2,3),(2,6),(4,6),(6,5)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [3,5,1,2,4,6,7] => ([(0,4),(1,3),(1,5),(3,6),(4,5),(5,6),(6,2)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [3,5,7,1,2,4,6] => ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,6,4,7] => [3,5,6,1,2,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,3,2,5,6,7,4] => [3,5,6,7,1,2,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,3,2,5,7,6,4] => [7,3,5,6,1,2,4] => ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,6,5,4,7] => [6,3,5,1,2,4,7] => ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(5,6)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,3,2,6,5,7,4] => [6,3,5,7,1,2,4] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,3,2,6,7,5,4] => [6,7,3,5,1,2,4] => ([(0,3),(1,5),(2,4),(2,6),(5,6)],7)
 => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,7,6,5,4] => [7,6,3,5,1,2,4] => ([(2,5),(3,4),(3,6),(5,6)],7)
 => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5,7,6] => [3,4,7,1,2,5,6] => ([(0,4),(1,5),(4,6),(5,2),(5,6),(6,3)],7)
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,3,4,2,6,5,7] => [3,4,6,1,2,5,7] => ([(0,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,7,5] => [3,4,6,7,1,2,5] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,7,6,5] => [7,3,4,6,1,2,5] => ([(1,3),(2,4),(3,6),(4,5),(4,6)],7)
 => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,3,4,5,2,7,6] => [3,4,5,7,1,2,6] => ([(0,5),(1,3),(3,6),(4,2),(4,6),(5,4)],7)
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,3,4,5,7,6,2] => [7,3,4,5,6,1,2] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,5,7,2] => [6,3,4,5,7,1,2] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,3,4,6,7,5,2] => [6,7,3,4,5,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,3,4,7,6,5,2] => [7,6,3,4,5,1,2] => ([(2,4),(3,5),(5,6)],7)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,3,5,4,2,7,6] => [5,3,4,7,1,2,6] => ([(0,5),(0,6),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,3,5,4,6,7,2] => [5,3,4,6,7,1,2] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,3,5,4,7,6,2] => [5,7,3,4,6,1,2] => ([(0,3),(1,5),(2,4),(2,6),(5,6)],7)
 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,3,5,6,4,7,2] => [5,6,3,4,7,1,2] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
 => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,5,6,7,4,2] => [5,6,7,3,4,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 1
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000834
(load all 4 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [[1,2]]
 => [1,2] => 1
[1,1,0,0]
 => [2,1] => [[1],[2]]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [[1,2,3]]
 => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => [3,1,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => [3,1,2] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [[1],[2],[3]]
 => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => [4,3,2,1] => 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] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [[1,2,3,5],[4,6],[7]]
 => [7,4,6,1,2,3,5] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,5,3,7] => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [[1,2,3,5],[4,6],[7]]
 => [7,4,6,1,2,3,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,4,3,7] => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => [[1,2,3,7],[4],[5],[6]]
 => [6,5,4,1,2,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => [[1,2,3,6],[4],[5],[7]]
 => [7,5,4,1,2,3,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,5,7,4,3] => [[1,2,3,5],[4],[6],[7]]
 => [7,6,4,1,2,3,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [[1,2,4,5],[3,6],[7]]
 => [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,3,2,5,6,7,4] => [[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,3,2,5,7,6,4] => [[1,2,4,5],[3,6],[7]]
 => [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,6,5,4,7] => [[1,2,4,7],[3,5],[6]]
 => [6,3,5,1,2,4,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,3,2,6,5,7,4] => [[1,2,4,6],[3,5],[7]]
 => [7,3,5,1,2,4,6] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,3,2,6,7,5,4] => [[1,2,4,5],[3,6],[7]]
 => [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,7,6,5,4] => [[1,2,4],[3,5],[6],[7]]
 => [7,6,3,5,1,2,4] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5,7,6] => [[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,3,4,2,6,7,5] => [[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,3,4,2,7,6,5] => [[1,2,3,5],[4,6],[7]]
 => [7,4,6,1,2,3,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,3,4,5,2,7,6] => [[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,3,4,6,5,2,7] => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,3,5,4,2,6,7] => [[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,3,5,4,2,7,6] => [[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,3,5,4,6,2,7] => [[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,3,5,4,7,6,2] => [[1,2,3,5],[4,6],[7]]
 => [7,4,6,1,2,3,5] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,3,5,6,4,2,7] => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,3,6,5,4,2,7] => [[1,2,3,7],[4],[5],[6]]
 => [6,5,4,1,2,3,7] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,6,5,4,7,2] => [[1,2,3,6],[4],[5],[7]]
 => [7,5,4,1,2,3,6] => ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,3,6,5,7,4,2] => [[1,2,3,5],[4],[6],[7]]
 => [7,6,4,1,2,3,5] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,4,3,2,5,6,7] => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,4,3,2,5,7,6] => [[1,2,5,6],[3,7],[4]]
 => [4,3,7,1,2,5,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,4,3,2,6,5,7] => [[1,2,5,7],[3,6],[4]]
 => [4,3,6,1,2,5,7] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,4,3,2,6,7,5] => [[1,2,5,6],[3,7],[4]]
 => [4,3,7,1,2,5,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,4,3,2,7,6,5] => [[1,2,5],[3,6],[4,7]]
 => [4,7,3,6,1,2,5] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,3,5,2,6,7] => [[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,4,3,5,2,7,6] => [[1,2,4,6],[3,7],[5]]
 => [5,3,7,1,2,4,6] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,3,5,6,2,7] => [[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,4,3,5,7,6,2] => [[1,2,4,5],[3,6],[7]]
 => [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,4,3,6,5,2,7] => [[1,2,4,7],[3,5],[6]]
 => [6,3,5,1,2,4,7] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,3,6,5,7,2] => [[1,2,4,6],[3,5],[7]]
 => [7,3,5,1,2,4,6] => ? = 2
Description
The number of right outer peaks of a permutation. A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$. In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000451
(load all 6 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [2,1] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,1,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,2,4,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,5,4,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [2,1,5,4,3] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,5,4,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,5,4,2] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,3,2,5,4] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,1,5,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,1,3,5,4] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,5,3,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,2,5,3,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,2,1,5,3] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,4,2,5,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,3,5,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,3,1,5,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,2,5,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,4,3,2,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,4,3,5,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,4,5,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [2,7,6,5,4,3,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [3,7,6,5,4,2,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [3,1,7,6,5,4,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [3,2,7,6,5,4,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [2,3,7,6,5,4,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,2,1] => [2,3,1,7,6,5,4] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [4,7,6,5,3,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [4,1,7,6,5,3,2] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [4,2,7,6,5,3,1] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,4,2,1] => [4,2,1,7,6,5,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [4,3,7,6,5,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,5,2,1] => [4,3,1,7,6,5,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,6,2,1] => [4,3,2,7,6,5,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,7,2,1] => [4,3,2,1,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [7,4,5,3,6,2,1] => [2,4,3,7,6,5,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,7,2,1] => [2,4,3,1,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [2,1,4,3,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [3,4,1,7,6,5,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [7,5,3,4,6,2,1] => [3,4,2,7,6,5,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => [3,4,2,1,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [3,1,4,2,7,6,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,4,3,5,6,2,1] => [3,2,4,7,6,5,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,7,2,1] => [3,2,4,1,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [3,2,1,4,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [2,3,4,7,6,5,1] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => [2,3,4,1,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [2,3,1,4,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [5,7,6,4,3,2,1] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [5,1,7,6,4,3,2] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [5,2,7,6,4,3,1] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,2,3,1] => [5,2,1,7,6,4,3] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,5,2,7,6,4,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [5,3,7,6,4,2,1] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [5,3,1,7,6,4,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,2,3,1] => [5,3,2,7,6,4,1] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,2,3,1] => [5,3,2,1,7,6,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [1,5,3,2,7,6,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [2,5,3,7,6,4,1] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,2,3,1] => [2,5,3,1,7,6,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,2,3,1] => [2,1,5,3,7,6,4] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,4,1] => [5,4,7,6,3,2,1] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,2,4,1] => [5,4,1,7,6,3,2] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,2,4,1] => [5,4,2,7,6,3,1] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,2,4,1] => [5,4,2,1,7,6,3] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,4,1] => [1,5,4,2,7,6,3] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,2,5,1] => [5,4,3,7,6,2,1] => ? = 1 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000455The second largest eigenvalue of a graph if it is integral. St000028The number of stack-sorts needed to sort a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001545The second Elser number of a connected graph. St000374The number of exclusive right-to-left minima of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St000628The balance of a binary word. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.