Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000498
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000498: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => 1
[2] => [1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 3
[1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
[2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 1
[3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 6
[1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,3},{2,4}}
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 3
[2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,4},{2,5},{3}}
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 5
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => {{1,3},{2,5},{4}}
 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => {{1,3},{2,4},{5}}
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,3,5},{2,4}}
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 6
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2,4},{3,5}}
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => {{1},{2},{3},{4},{5},{6}}
 => 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => {{1,6},{2},{3},{4},{5}}
 => 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3},{4,5},{6}}
 => {{1,5},{2},{3},{4},{6}}
 => 7
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3},{4,5,6}}
 => {{1,5},{2,6},{3},{4}}
 => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => {{1},{2},{3,4},{5},{6}}
 => {{1,4},{2},{3},{5},{6}}
 => 8
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => {{1},{2},{3,4},{5,6}}
 => {{1,4},{2,6},{3},{5}}
 => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => {{1},{2},{3,4,5},{6}}
 => {{1,4},{2,5},{3},{6}}
 => 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => {{1,4},{2,5},{3,6}}
 => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => {{1},{2,3},{4},{5},{6}}
 => {{1,3},{2},{4},{5},{6}}
 => 9
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => {{1},{2,3},{4},{5,6}}
 => {{1,3},{2,6},{4},{5}}
 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1},{2,3},{4,5},{6}}
 => {{1,3},{2,5},{4},{6}}
 => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => {{1},{2,3},{4,5,6}}
 => {{1,3,6},{2,5},{4}}
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6}}
 => {{1,3},{2,4},{5},{6}}
 => 6
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => {{1,3,6},{2,4},{5}}
 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3,4,5},{6}}
 => {{1,3,5},{2,4},{6}}
 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => {{1,3,5},{2,4,6}}
 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => {{1,2},{3},{4},{5},{6}}
 => 10
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => {{1,2,6},{3},{4},{5}}
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => {{1,2,5},{3},{4},{6}}
 => 4
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => {{1,2,5},{3,6},{4}}
 => 1
Description
The lcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''lcs''' (left-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.