Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000502
(load all 6 compositions to match this statistic)
Mapping
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 1
{{1},{2}}
 => 0
{{1,2,3}}
 => 2
{{1,2},{3}}
 => 1
{{1,3},{2}}
 => 0
{{1},{2,3}}
 => 1
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 3
{{1,2,3},{4}}
 => 2
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 2
{{1,2},{3},{4}}
 => 1
{{1,3,4},{2}}
 => 1
{{1,3},{2,4}}
 => 0
{{1,3},{2},{4}}
 => 0
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 2
{{1},{2,3},{4}}
 => 1
{{1,4},{2},{3}}
 => 0
{{1},{2,4},{3}}
 => 0
{{1},{2},{3,4}}
 => 1
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 4
{{1,2,3,4},{5}}
 => 3
{{1,2,3,5},{4}}
 => 2
{{1,2,3},{4,5}}
 => 3
{{1,2,3},{4},{5}}
 => 2
{{1,2,4,5},{3}}
 => 2
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 2
{{1,2},{3,4,5}}
 => 3
{{1,2},{3,4},{5}}
 => 2
{{1,2,5},{3},{4}}
 => 1
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 2
{{1,2},{3},{4},{5}}
 => 1
{{1,3,4,5},{2}}
 => 2
{{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 0
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 0
{{1,3,5},{2},{4}}
 => 0
{{1,3},{2,5},{4}}
 => 0
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 0
{{1,4,5},{2,3}}
 => 2
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 1
Description
The number of successions of a set partitions. This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000382
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [2] => 2 = 1 + 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => 1 = 0 + 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => 3 = 2 + 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => 2 = 1 + 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => 1 = 0 + 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => 2 = 1 + 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => 1 = 0 + 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => 4 = 3 + 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => 3 = 2 + 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => 2 = 1 + 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => 3 = 2 + 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => 2 = 1 + 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => 2 = 1 + 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => 1 = 0 + 1
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => 1 = 0 + 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => 2 = 1 + 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => 2 = 1 + 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => 1 = 0 + 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => 1 = 0 + 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => 2 = 1 + 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => 1 = 0 + 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => 5 = 4 + 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => 4 = 3 + 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => 3 = 2 + 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => 3 = 2 + 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => 3 = 2 + 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => 2 = 1 + 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => 3 = 2 + 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => 2 = 1 + 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => 3 = 2 + 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => 2 = 1 + 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => 1 = 0 + 1
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => 1 = 0 + 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => 3 = 2 + 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => 2 = 1 + 1
{{1,4,5,6,7,8},{2},{3}}
 => {{1,5,6,7,8},{2},{3,4}}
 => ? => ? = 4 + 1
{{1,3,4,5,6,8},{2,7}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? => ? = 3 + 1
{{1,3,4,5,6,7},{2,8}}
 => {{1,4,5,6,7},{2,3,8}}
 => ? => ? = 4 + 1
{{1,6},{2,3,4,5,7,8}}
 => {{1,3,4,5,8},{2,6,7}}
 => ? => ? = 4 + 1
{{1,7},{2,3,4,5,6,8}}
 => {{1,3,4,5,6},{2,7,8}}
 => ? => ? = 4 + 1
Description
The first part of an integer composition.
Matching statistic: St000439
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [2] => [1,1,0,0]
 => 3 = 1 + 2
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [1,0,1,0]
 => 2 = 0 + 2
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 4 = 2 + 2
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 1 + 2
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 1 + 2
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 0 + 2
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 2 + 2
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 2 + 2
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 2 + 2
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 1 + 2
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5 + 2
{{1},{2,4,5,6,7,8},{3}}
 => {{1,5,6,7,8},{2,4},{3}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 2
{{1},{2,3,5,6,7,8},{4}}
 => {{1,3,6,7,8},{2,5},{4}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 2
{{1},{2,3,4,6,7,8},{5}}
 => {{1,3,4,7,8},{2,6},{5}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 2
{{1},{2,3,4,5,7,8},{6}}
 => {{1,3,4,5,8},{2,7},{6}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 2
{{1},{2,3,4,5,6,7},{8}}
 => {{1,3,4,5,6,7},{2},{8}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5 + 2
{{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6},{2,8},{7}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 2
{{1,4,5,6,7,8},{2},{3}}
 => {{1,5,6,7,8},{2},{3,4}}
 => ? => ?
 => ? = 4 + 2
{{1,3,5,6,7,8},{2},{4}}
 => {{1,6,7,8},{2,3,5},{4}}
 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 2
{{1,3,4,5,6,8},{2,7}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? => ?
 => ? = 3 + 2
{{1,3,4,5,6,7},{2,8}}
 => {{1,4,5,6,7},{2,3,8}}
 => ? => ?
 => ? = 4 + 2
{{1,2,4,5,6,7},{3,8}}
 => {{1,2,5,6,7},{3,4,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
{{1,2,3,5,6,7},{4,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
{{1,2,3,4,6,7},{5,8}}
 => {{1,2,3,4,7},{5,6,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
{{1,2,3,4,5,7},{6,8}}
 => {{1,2,3,4,5},{6,7,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
{{1,3},{2,4,5,6,7,8}}
 => {{1,5,6,7,8},{2,3,4}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
{{1,4},{2,3,5,6,7,8}}
 => {{1,3,6,7,8},{2,4,5}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
{{1,5},{2,3,4,6,7,8}}
 => {{1,3,4,7,8},{2,5,6}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
{{1,6},{2,3,4,5,7,8}}
 => {{1,3,4,5,8},{2,6,7}}
 => ? => ?
 => ? = 4 + 2
{{1,7},{2,3,4,5,6,8}}
 => {{1,3,4,5,6},{2,7,8}}
 => ? => ?
 => ? = 4 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000504
(load all 3 compositions to match this statistic)
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => 2 = 1 + 1
{{1},{2}}
 => {{1},{2}}
 => 1 = 0 + 1
{{1,2,3}}
 => {{1,2,3}}
 => 3 = 2 + 1
{{1,2},{3}}
 => {{1,2},{3}}
 => 2 = 1 + 1
{{1,3},{2}}
 => {{1},{2,3}}
 => 1 = 0 + 1
{{1},{2,3}}
 => {{1,3},{2}}
 => 2 = 1 + 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 1 = 0 + 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 4 = 3 + 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 3 = 2 + 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 2 = 1 + 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 3 = 2 + 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 2 = 1 + 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 2 = 1 + 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 1 = 0 + 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 2 = 1 + 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 3 = 2 + 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 5 = 4 + 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 4 = 3 + 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => 3 = 2 + 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => 3 = 2 + 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => 3 = 2 + 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => 3 = 2 + 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1 = 0 + 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => 3 = 2 + 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => 2 = 1 + 1
{{1},{2,4,5,6,7,8},{3}}
 => {{1,5,6,7,8},{2,4},{3}}
 => ? = 4 + 1
{{1},{2,3,5,6,7,8},{4}}
 => {{1,3,6,7,8},{2,5},{4}}
 => ? = 4 + 1
{{1},{2,3,4,6,7,8},{5}}
 => {{1,3,4,7,8},{2,6},{5}}
 => ? = 4 + 1
{{1},{2,3,4,5,7,8},{6}}
 => {{1,3,4,5,8},{2,7},{6}}
 => ? = 4 + 1
{{1},{2,3,4,5,6,7},{8}}
 => {{1,3,4,5,6,7},{2},{8}}
 => ? = 5 + 1
{{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6},{2,8},{7}}
 => ? = 4 + 1
{{1,4,5,6,7,8},{2},{3}}
 => {{1,5,6,7,8},{2},{3,4}}
 => ? = 4 + 1
{{1,3,5,6,7,8},{2},{4}}
 => {{1,6,7,8},{2,3,5},{4}}
 => ? = 3 + 1
{{1,3,5,6,7,8},{2,4}}
 => {{1,6,7,8},{2,3,4,5}}
 => ? = 3 + 1
{{1,3,4,6,7,8},{2,5}}
 => {{1,4,7,8},{2,3,5,6}}
 => ? = 3 + 1
{{1,2,4,6,7,8},{3,5}}
 => {{1,2,7,8},{3,4,5,6}}
 => ? = 3 + 1
{{1,3,4,5,7,8},{2,6}}
 => {{1,4,5,8},{2,3,6,7}}
 => ? = 3 + 1
{{1,2,4,5,7,8},{3,6}}
 => {{1,2,5,8},{3,4,6,7}}
 => ? = 3 + 1
{{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,8},{4,5,6,7}}
 => ? = 3 + 1
{{1,3,4,5,6,8},{2,7}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? = 3 + 1
{{1,2,4,5,6,8},{3,7}}
 => {{1,2,5,6},{3,4,7,8}}
 => ? = 3 + 1
{{1,2,3,5,6,8},{4,7}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 3 + 1
{{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 3 + 1
{{1,3,4,5,6,7},{2,8}}
 => {{1,4,5,6,7},{2,3,8}}
 => ? = 4 + 1
{{1,2,4,5,6,7},{3,8}}
 => {{1,2,5,6,7},{3,4,8}}
 => ? = 4 + 1
{{1,2,3,5,6,7},{4,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => ? = 4 + 1
{{1,2,3,4,6,7},{5,8}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 4 + 1
{{1,2,3,4,5,7},{6,8}}
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 4 + 1
{{1,3},{2,4,5,6,7,8}}
 => {{1,5,6,7,8},{2,3,4}}
 => ? = 4 + 1
{{1,4},{2,3,5,6,7,8}}
 => {{1,3,6,7,8},{2,4,5}}
 => ? = 4 + 1
{{1,5},{2,3,4,6,7,8}}
 => {{1,3,4,7,8},{2,5,6}}
 => ? = 4 + 1
{{1,6},{2,3,4,5,7,8}}
 => {{1,3,4,5,8},{2,6,7}}
 => ? = 4 + 1
{{1,7},{2,3,4,5,6,8}}
 => {{1,3,4,5,6},{2,7,8}}
 => ? = 4 + 1
Description
The cardinality of the first block of a set partition. The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000383
(load all 2 compositions to match this statistic)
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
{{1,2}}
 => {{1,2}}
 => [2] => [2] => 2 = 1 + 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [1,1] => 1 = 0 + 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [3] => 3 = 2 + 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,2] => 2 = 1 + 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [2,1] => 1 = 0 + 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,2] => 2 = 1 + 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [1,1,1] => 1 = 0 + 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [4] => 4 = 3 + 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,3] => 3 = 2 + 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [2,2] => 2 = 1 + 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,3] => 3 = 2 + 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,1,2] => 2 = 1 + 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => [2,2] => 2 = 1 + 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => [3,1] => 1 = 0 + 1
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [1,2,1] => 1 = 0 + 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [2,2] => 2 = 1 + 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,3] => 3 = 2 + 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,1,2] => 2 = 1 + 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [2,1,1] => 1 = 0 + 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [1,2,1] => 1 = 0 + 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,1,2] => 2 = 1 + 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [5] => 5 = 4 + 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,4] => 4 = 3 + 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [1,4] => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [3,2] => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [1,4] => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [2,1,2] => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,1,2] => 2 = 1 + 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => [3,2] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => [4,1] => 1 = 0 + 1
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [3,2] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [1,3,1] => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [1,3,1] => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [2,2,1] => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,1,2,1] => 1 = 0 + 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [3,2] => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => [6,1,1] => [1,1,6] => ? = 5 + 1
{{1},{2,4,5,6,7,8},{3}}
 => {{1,5,6,7,8},{2,4},{3}}
 => [5,2,1] => [1,2,5] => ? = 4 + 1
{{1},{2,3,5,6,7,8},{4}}
 => {{1,3,6,7,8},{2,5},{4}}
 => [5,2,1] => [1,2,5] => ? = 4 + 1
{{1},{2,3,4,6,7,8},{5}}
 => {{1,3,4,7,8},{2,6},{5}}
 => [5,2,1] => [1,2,5] => ? = 4 + 1
{{1},{2,3,4,5,7,8},{6}}
 => {{1,3,4,5,8},{2,7},{6}}
 => [5,2,1] => [1,2,5] => ? = 4 + 1
{{1},{2,3,4,5,6,7},{8}}
 => {{1,3,4,5,6,7},{2},{8}}
 => [6,1,1] => [1,1,6] => ? = 5 + 1
{{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6},{2,8},{7}}
 => [5,2,1] => [1,2,5] => ? = 4 + 1
{{1,4,5,6,7,8},{2},{3}}
 => {{1,5,6,7,8},{2},{3,4}}
 => ? => ? => ? = 4 + 1
{{1,3,5,6,7,8},{2},{4}}
 => {{1,6,7,8},{2,3,5},{4}}
 => [4,3,1] => [1,3,4] => ? = 3 + 1
{{1,3,4,5,6,7,8},{2}}
 => {{1,4,5,6,7,8},{2,3}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,4,5,6,7,8},{2,3}}
 => {{1,3,5,6,7,8},{2,4}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,4,5,6,7,8},{3}}
 => {{1,2,5,6,7,8},{3,4}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,5,6,7,8},{3,4}}
 => {{1,2,4,6,7,8},{3,5}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,5,6,7,8},{4}}
 => {{1,2,3,6,7,8},{4,5}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,6,7,8},{4,5}}
 => {{1,2,3,5,7,8},{4,6}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,4,6,7,8},{5}}
 => {{1,2,3,4,7,8},{5,6}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,4,7,8},{5,6}}
 => {{1,2,3,4,6,8},{5,7}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,4,5,7,8},{6}}
 => {{1,2,3,4,5,8},{6,7}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,8},{2,3,4,5,6,7}}
 => {{1,3,4,5,6,7},{2,8}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,4,5,8},{6,7}}
 => {{1,2,3,4,5,7},{6,8}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,4,5,6,8},{7}}
 => {{1,2,3,4,5,6},{7,8}}
 => [6,2] => [2,6] => ? = 5 + 1
{{1,2,3,4,5,6,7,8}}
 => {{1,2,3,4,5,6,7,8}}
 => [8] => [8] => ? = 7 + 1
{{1,3,4,5,6,8},{2,7}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? => ? => ? = 3 + 1
{{1,3,4,5,6,7},{2,8}}
 => {{1,4,5,6,7},{2,3,8}}
 => ? => ? => ? = 4 + 1
{{1,2,4,5,6,7},{3,8}}
 => {{1,2,5,6,7},{3,4,8}}
 => [5,3] => [3,5] => ? = 4 + 1
{{1,2,3,5,6,7},{4,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => [5,3] => [3,5] => ? = 4 + 1
{{1,2,3,4,6,7},{5,8}}
 => {{1,2,3,4,7},{5,6,8}}
 => [5,3] => [3,5] => ? = 4 + 1
{{1,2,3,4,5,7},{6,8}}
 => {{1,2,3,4,5},{6,7,8}}
 => [5,3] => [3,5] => ? = 4 + 1
{{1,3},{2,4,5,6,7,8}}
 => {{1,5,6,7,8},{2,3,4}}
 => [5,3] => [3,5] => ? = 4 + 1
{{1,4},{2,3,5,6,7,8}}
 => {{1,3,6,7,8},{2,4,5}}
 => [5,3] => [3,5] => ? = 4 + 1
{{1,5},{2,3,4,6,7,8}}
 => {{1,3,4,7,8},{2,5,6}}
 => [5,3] => [3,5] => ? = 4 + 1
{{1,6},{2,3,4,5,7,8}}
 => {{1,3,4,5,8},{2,6,7}}
 => ? => ? => ? = 4 + 1
{{1,7},{2,3,4,5,6,8}}
 => {{1,3,4,5,6},{2,7,8}}
 => ? => ? => ? = 4 + 1
Description
The last part of an integer composition.
Matching statistic: St000025
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 96%●distinct values known / distinct values provided: 88%
Values
{{1,2}}
 => {{1,2}}
 => [2] => [1,1,0,0]
 => 2 = 1 + 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5 + 1
{{1},{2,4,5,6,7,8},{3}}
 => {{1,5,6,7,8},{2,4},{3}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,5,6,7,8},{4}}
 => {{1,3,6,7,8},{2,5},{4}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,6,7,8},{5}}
 => {{1,3,4,7,8},{2,6},{5}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,5,7,8},{6}}
 => {{1,3,4,5,8},{2,7},{6}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,5,6,7},{8}}
 => {{1,3,4,5,6,7},{2},{8}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5 + 1
{{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6},{2,8},{7}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,5,6,7,8}}
 => {{1,3,4,5,6,7,8},{2}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,2},{3,4,5,6,7,8}}
 => {{1,2,4,5,6,7,8},{3}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,4,5,6,7,8},{2},{3}}
 => {{1,5,6,7,8},{2},{3,4}}
 => ? => ?
 => ? = 4 + 1
{{1,3,5,6,7,8},{2},{4}}
 => {{1,6,7,8},{2,3,5},{4}}
 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
{{1,3,4,5,6,7,8},{2}}
 => {{1,4,5,6,7,8},{2,3}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,4,5,6,7,8},{2,3}}
 => {{1,3,5,6,7,8},{2,4}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,4,5,6,7,8},{3}}
 => {{1,2,5,6,7,8},{3,4}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,5,6,7,8},{3,4}}
 => {{1,2,4,6,7,8},{3,5}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,5,6,7,8},{4}}
 => {{1,2,3,6,7,8},{4,5}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,6,7,8},{4,5}}
 => {{1,2,3,5,7,8},{4,6}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,6,7,8},{5}}
 => {{1,2,3,4,7,8},{5,6}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6},{7,8}}
 => {{1,2,3,4,5,6,8},{7}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,2,3,4,7,8},{5,6}}
 => {{1,2,3,4,6,8},{5,7}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,7,8},{6}}
 => {{1,2,3,4,5,8},{6,7}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6,7},{8}}
 => {{1,2,3,4,5,6,7},{8}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,8},{2,3,4,5,6,7}}
 => {{1,3,4,5,6,7},{2,8}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,8},{6,7}}
 => {{1,2,3,4,5,7},{6,8}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6,8},{7}}
 => {{1,2,3,4,5,6},{7,8}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6,7,8}}
 => {{1,2,3,4,5,6,7,8}}
 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
{{1,3,5,6,7,8},{2,4}}
 => {{1,6,7,8},{2,3,4,5}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,6,7,8},{2,5}}
 => {{1,4,7,8},{2,3,5,6}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,4,6,7,8},{3,5}}
 => {{1,2,7,8},{3,4,5,6}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,5,7,8},{2,6}}
 => {{1,4,5,8},{2,3,6,7}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,4,5,7,8},{3,6}}
 => {{1,2,5,8},{3,4,6,7}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,8},{4,5,6,7}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,5,6,8},{2,7}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? => ?
 => ? = 3 + 1
{{1,2,4,5,6,8},{3,7}}
 => {{1,2,5,6},{3,4,7,8}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,3,5,6,8},{4,7}}
 => {{1,2,3,6},{4,5,7,8}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4},{5,6,7,8}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,5,6,7},{2,8}}
 => {{1,4,5,6,7},{2,3,8}}
 => ? => ?
 => ? = 4 + 1
{{1,2,4,5,6,7},{3,8}}
 => {{1,2,5,6,7},{3,4,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,2,3,5,6,7},{4,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,2,3,4,6,7},{5,8}}
 => {{1,2,3,4,7},{5,6,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,2,3,4,5,7},{6,8}}
 => {{1,2,3,4,5},{6,7,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,3},{2,4,5,6,7,8}}
 => {{1,5,6,7,8},{2,3,4}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,4},{2,3,5,6,7,8}}
 => {{1,3,6,7,8},{2,4,5}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,5},{2,3,4,6,7,8}}
 => {{1,3,4,7,8},{2,5,6}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,6},{2,3,4,5,7,8}}
 => {{1,3,4,5,8},{2,6,7}}
 => ? => ?
 => ? = 4 + 1
{{1,7},{2,3,4,5,6,8}}
 => {{1,3,4,5,6},{2,7,8}}
 => ? => ?
 => ? = 4 + 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 96%●distinct values known / distinct values provided: 88%
Values
{{1,2}}
 => {{1,2}}
 => [2] => [1,1,0,0]
 => 2 = 1 + 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5 + 1
{{1},{2,4,5,6,7,8},{3}}
 => {{1,5,6,7,8},{2,4},{3}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,5,6,7,8},{4}}
 => {{1,3,6,7,8},{2,5},{4}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,6,7,8},{5}}
 => {{1,3,4,7,8},{2,6},{5}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,5,7,8},{6}}
 => {{1,3,4,5,8},{2,7},{6}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,5,6,7},{8}}
 => {{1,3,4,5,6,7},{2},{8}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5 + 1
{{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6},{2,8},{7}}
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
{{1},{2,3,4,5,6,7,8}}
 => {{1,3,4,5,6,7,8},{2}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,2},{3,4,5,6,7,8}}
 => {{1,2,4,5,6,7,8},{3}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,4,5,6,7,8},{2},{3}}
 => {{1,5,6,7,8},{2},{3,4}}
 => ? => ?
 => ? = 4 + 1
{{1,3,5,6,7,8},{2},{4}}
 => {{1,6,7,8},{2,3,5},{4}}
 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
{{1,3,4,5,6,7,8},{2}}
 => {{1,4,5,6,7,8},{2,3}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,4,5,6,7,8},{2,3}}
 => {{1,3,5,6,7,8},{2,4}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,4,5,6,7,8},{3}}
 => {{1,2,5,6,7,8},{3,4}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,5,6,7,8},{3,4}}
 => {{1,2,4,6,7,8},{3,5}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,5,6,7,8},{4}}
 => {{1,2,3,6,7,8},{4,5}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,6,7,8},{4,5}}
 => {{1,2,3,5,7,8},{4,6}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,6,7,8},{5}}
 => {{1,2,3,4,7,8},{5,6}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6},{7,8}}
 => {{1,2,3,4,5,6,8},{7}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,2,3,4,7,8},{5,6}}
 => {{1,2,3,4,6,8},{5,7}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,7,8},{6}}
 => {{1,2,3,4,5,8},{6,7}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6,7},{8}}
 => {{1,2,3,4,5,6,7},{8}}
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6 + 1
{{1,8},{2,3,4,5,6,7}}
 => {{1,3,4,5,6,7},{2,8}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,8},{6,7}}
 => {{1,2,3,4,5,7},{6,8}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6,8},{7}}
 => {{1,2,3,4,5,6},{7,8}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
{{1,2,3,4,5,6,7,8}}
 => {{1,2,3,4,5,6,7,8}}
 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
{{1,3,5,6,7,8},{2,4}}
 => {{1,6,7,8},{2,3,4,5}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,6,7,8},{2,5}}
 => {{1,4,7,8},{2,3,5,6}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,4,6,7,8},{3,5}}
 => {{1,2,7,8},{3,4,5,6}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,5,7,8},{2,6}}
 => {{1,4,5,8},{2,3,6,7}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,4,5,7,8},{3,6}}
 => {{1,2,5,8},{3,4,6,7}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,8},{4,5,6,7}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,5,6,8},{2,7}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? => ?
 => ? = 3 + 1
{{1,2,4,5,6,8},{3,7}}
 => {{1,2,5,6},{3,4,7,8}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,3,5,6,8},{4,7}}
 => {{1,2,3,6},{4,5,7,8}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4},{5,6,7,8}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
{{1,3,4,5,6,7},{2,8}}
 => {{1,4,5,6,7},{2,3,8}}
 => ? => ?
 => ? = 4 + 1
{{1,2,4,5,6,7},{3,8}}
 => {{1,2,5,6,7},{3,4,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,2,3,5,6,7},{4,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,2,3,4,6,7},{5,8}}
 => {{1,2,3,4,7},{5,6,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,2,3,4,5,7},{6,8}}
 => {{1,2,3,4,5},{6,7,8}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,3},{2,4,5,6,7,8}}
 => {{1,5,6,7,8},{2,3,4}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,4},{2,3,5,6,7,8}}
 => {{1,3,6,7,8},{2,4,5}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,5},{2,3,4,6,7,8}}
 => {{1,3,4,7,8},{2,5,6}}
 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
{{1,6},{2,3,4,5,7,8}}
 => {{1,3,4,5,8},{2,6,7}}
 => ? => ?
 => ? = 4 + 1
{{1,7},{2,3,4,5,6,8}}
 => {{1,3,4,5,6},{2,7,8}}
 => ? => ?
 => ? = 4 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000237
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => 1
{{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => 2
{{1,2},{3}}
 => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => 3
{{1,2,3},{4}}
 => [2,3,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 4
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 1
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => ? = 4
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => ? = 4
{{1,2,3,4,6},{5},{7}}
 => [2,3,4,6,5,1,7] => ? = 3
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => ? = 4
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => ? = 5
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => ? = 4
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => ? = 3
{{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => ? = 3
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => ? = 4
{{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => ? = 3
{{1,2,3,5,6},{4},{7}}
 => [2,3,5,4,6,1,7] => ? = 3
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => ? = 2
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ? = 3
{{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => ? = 2
{{1,2,3,5,7},{4},{6}}
 => [2,3,5,4,7,6,1] => ? = 2
{{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => ? = 2
{{1,2,3,5},{4},{6,7}}
 => [2,3,5,4,1,7,6] => ? = 3
{{1,2,3,5},{4},{6},{7}}
 => [2,3,5,4,1,6,7] => ? = 2
{{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => ? = 4
{{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => ? = 3
{{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => ? = 3
{{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => ? = 4
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => ? = 5
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => ? = 4
{{1,2,3,7},{4,5},{6}}
 => [2,3,7,5,4,6,1] => ? = 3
{{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => ? = 3
{{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => ? = 4
{{1,2,3},{4,5},{6},{7}}
 => [2,3,1,5,4,6,7] => ? = 3
{{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => ? = 3
{{1,2,3,6},{4,7},{5}}
 => [2,3,6,7,5,1,4] => ? = 2
{{1,2,3,6},{4},{5,7}}
 => [2,3,6,4,7,1,5] => ? = 2
{{1,2,3,6},{4},{5},{7}}
 => [2,3,6,4,5,1,7] => ? = 2
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => ? = 2
{{1,2,3},{4,6,7},{5}}
 => [2,3,1,6,5,7,4] => ? = 3
{{1,2,3},{4,6},{5,7}}
 => [2,3,1,6,7,4,5] => ? = 2
{{1,2,3},{4,6},{5},{7}}
 => [2,3,1,6,5,4,7] => ? = 2
{{1,2,3,7},{4},{5,6}}
 => [2,3,7,4,6,5,1] => ? = 3
{{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => ? = 3
{{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => ? = 4
{{1,2,3},{4},{5,6},{7}}
 => [2,3,1,4,6,5,7] => ? = 3
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => ? = 2
{{1,2,3},{4,7},{5},{6}}
 => [2,3,1,7,5,6,4] => ? = 2
{{1,2,3},{4},{5,7},{6}}
 => [2,3,1,4,7,6,5] => ? = 2
{{1,2,3},{4},{5},{6,7}}
 => [2,3,1,4,5,7,6] => ? = 3
{{1,2,4,5,6,7},{3}}
 => [2,4,3,5,6,7,1] => ? = 4
{{1,2,4,5,6},{3,7}}
 => [2,4,7,5,6,1,3] => ? = 3
{{1,2,4,5,6},{3},{7}}
 => [2,4,3,5,6,1,7] => ? = 3
{{1,2,4,5,7},{3,6}}
 => [2,4,6,5,7,3,1] => ? = 2
{{1,2,4,5},{3,6,7}}
 => [2,4,6,5,1,7,3] => ? = 3
{{1,2,4,5},{3,6},{7}}
 => [2,4,6,5,1,3,7] => ? = 2
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000214
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [3,2,1] => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 3
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 4
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => [6,7,5,4,3,2,1] => ? = 4
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => [5,7,6,4,3,2,1] => ? = 4
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => [6,4,3,2,1,7,5] => ? = 3
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => [6,5,7,4,3,2,1] => ? = 4
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => [4,3,2,1,6,5,7] => ? = 4
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => [5,6,7,4,3,2,1] => ? = 3
{{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => [4,1,2,3,7,6,5] => [4,3,2,1,6,7,5] => ? = 3
{{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 4
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => [4,7,6,5,3,2,1] => ? = 4
{{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => [6,1,2,7,3,5,4] => [6,5,3,2,1,7,4] => ? = 3
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => ? = 2
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [5,1,2,7,3,4,6] => [5,3,2,1,7,6,4] => ? = 3
{{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => [5,1,2,6,3,4,7] => [5,3,2,1,6,4,7] => ? = 2
{{1,2,3,5,7},{4},{6}}
 => [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => [4,6,7,5,3,2,1] => ? = 2
{{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => [5,1,2,7,3,6,4] => [5,3,2,1,6,7,4] => ? = 2
{{1,2,3,5},{4},{6,7}}
 => [2,3,5,4,1,7,6] => [5,1,2,4,3,7,6] => [4,5,3,2,1,7,6] => ? = 3
{{1,2,3,5},{4},{6},{7}}
 => [2,3,5,4,1,6,7] => [5,1,2,4,3,6,7] => [4,5,3,2,1,6,7] => ? = 2
{{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => [7,1,2,5,4,3,6] => [5,4,7,6,3,2,1] => ? = 4
{{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => [6,1,2,7,4,3,5] => [6,3,2,1,7,5,4] => ? = 3
{{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => [7,1,2,6,4,5,3] => [6,5,4,7,3,2,1] => ? = 4
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => [3,2,1,7,6,5,4] => ? = 5
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => [3,2,1,6,5,4,7] => ? = 4
{{1,2,3,7},{4,5},{6}}
 => [2,3,7,5,4,6,1] => [7,1,2,5,4,6,3] => [5,4,6,7,3,2,1] => ? = 3
{{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => [3,1,2,7,4,6,5] => [3,2,1,6,7,5,4] => ? = 3
{{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 4
{{1,2,3},{4,5},{6},{7}}
 => [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 3
{{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => [4,5,7,6,3,2,1] => ? = 3
{{1,2,3,6},{4,7},{5}}
 => [2,3,6,7,5,1,4] => [6,1,2,7,5,3,4] => [5,6,3,2,1,7,4] => ? = 2
{{1,2,3,6},{4},{5,7}}
 => [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => [4,6,3,2,1,7,5] => ? = 2
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => [5,6,4,7,3,2,1] => ? = 2
{{1,2,3},{4,6,7},{5}}
 => [2,3,1,6,5,7,4] => [3,1,2,7,5,4,6] => [3,2,1,5,7,6,4] => ? = 3
{{1,2,3},{4,6},{5,7}}
 => [2,3,1,6,7,4,5] => [3,1,2,6,7,4,5] => [3,2,1,6,4,7,5] => ? = 2
{{1,2,3},{4,6},{5},{7}}
 => [2,3,1,6,5,4,7] => [3,1,2,6,5,4,7] => [3,2,1,5,6,4,7] => ? = 2
{{1,2,3,7},{4},{5,6}}
 => [2,3,7,4,6,5,1] => [7,1,2,4,6,5,3] => [4,6,5,7,3,2,1] => ? = 3
{{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => [3,1,2,7,6,5,4] => [3,2,1,6,5,7,4] => ? = 3
{{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => [3,2,1,4,7,6,5] => ? = 4
{{1,2,3},{4},{5,6},{7}}
 => [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 3
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => [4,5,6,7,3,2,1] => ? = 2
{{1,2,3},{4,7},{5},{6}}
 => [2,3,1,7,5,6,4] => [3,1,2,7,5,6,4] => [3,2,1,5,6,7,4] => ? = 2
{{1,2,3},{4},{5,7},{6}}
 => [2,3,1,4,7,6,5] => [3,1,2,4,7,6,5] => [3,2,1,4,6,7,5] => ? = 2
{{1,2,3},{4},{5},{6,7}}
 => [2,3,1,4,5,7,6] => [3,1,2,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 3
{{1,2,4,5,6,7},{3}}
 => [2,4,3,5,6,7,1] => [7,1,3,2,4,5,6] => [3,7,6,5,4,2,1] => ? = 4
{{1,2,4,5,6},{3,7}}
 => [2,4,7,5,6,1,3] => [6,1,7,2,4,5,3] => [6,5,4,2,1,7,3] => ? = 3
{{1,2,4,5,7},{3,6}}
 => [2,4,6,5,7,3,1] => [7,1,6,2,4,3,5] => [6,3,7,5,4,2,1] => ? = 2
{{1,2,4,5},{3,6,7}}
 => [2,4,6,5,1,7,3] => [5,1,7,2,4,3,6] => [5,4,2,1,7,6,3] => ? = 3
{{1,2,4,5},{3,6},{7}}
 => [2,4,6,5,1,3,7] => [5,1,6,2,4,3,7] => [5,4,2,1,6,3,7] => ? = 2
{{1,2,4,5,7},{3},{6}}
 => [2,4,3,5,7,6,1] => [7,1,3,2,4,6,5] => [3,6,7,5,4,2,1] => ? = 2
{{1,2,4,5},{3,7},{6}}
 => [2,4,7,5,1,6,3] => [5,1,7,2,4,6,3] => [5,4,2,1,6,7,3] => ? = 2
{{1,2,4,5},{3},{6,7}}
 => [2,4,3,5,1,7,6] => [5,1,3,2,4,7,6] => [3,5,4,2,1,7,6] => ? = 3
Description
The number of adjacencies of a permutation. An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].