Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001092
(load all 3 compositions to match this statistic)
Mapping
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0 = 1 - 1
[2]
 => 1 = 2 - 1
[1,1]
 => 0 = 1 - 1
[3]
 => 0 = 1 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 0 = 1 - 1
[4]
 => 1 = 2 - 1
[3,1]
 => 0 = 1 - 1
[2,2]
 => 1 = 2 - 1
[2,1,1]
 => 1 = 2 - 1
[1,1,1,1]
 => 0 = 1 - 1
[5]
 => 0 = 1 - 1
[4,1]
 => 1 = 2 - 1
[3,2]
 => 1 = 2 - 1
[3,1,1]
 => 0 = 1 - 1
[2,2,1]
 => 1 = 2 - 1
[2,1,1,1]
 => 1 = 2 - 1
[1,1,1,1,1]
 => 0 = 1 - 1
[5,1]
 => 0 = 1 - 1
[4,2]
 => 2 = 3 - 1
[4,1,1]
 => 1 = 2 - 1
[3,3]
 => 0 = 1 - 1
[3,2,1]
 => 1 = 2 - 1
[3,1,1,1]
 => 0 = 1 - 1
[2,2,2]
 => 1 = 2 - 1
[2,2,1,1]
 => 1 = 2 - 1
[2,1,1,1,1]
 => 1 = 2 - 1
[5,2]
 => 1 = 2 - 1
[5,1,1]
 => 0 = 1 - 1
[4,3]
 => 1 = 2 - 1
[4,2,1]
 => 2 = 3 - 1
[4,1,1,1]
 => 1 = 2 - 1
[3,3,1]
 => 0 = 1 - 1
[3,2,2]
 => 1 = 2 - 1
[3,2,1,1]
 => 1 = 2 - 1
[3,1,1,1,1]
 => 0 = 1 - 1
[2,2,2,1]
 => 1 = 2 - 1
[2,2,1,1,1]
 => 1 = 2 - 1
[5,3]
 => 0 = 1 - 1
[5,2,1]
 => 1 = 2 - 1
[5,1,1,1]
 => 0 = 1 - 1
[4,4]
 => 1 = 2 - 1
[4,3,1]
 => 1 = 2 - 1
[4,2,2]
 => 2 = 3 - 1
[4,2,1,1]
 => 2 = 3 - 1
[4,1,1,1,1]
 => 1 = 2 - 1
[3,3,2]
 => 1 = 2 - 1
[3,3,1,1]
 => 0 = 1 - 1
[3,2,2,1]
 => 1 = 2 - 1
[3,2,1,1,1]
 => 1 = 2 - 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St001151
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => {{1,3},{2}}
 => 1
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}}
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => {{1,5},{2,3},{4}}
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => {{1,5},{2},{3,4}}
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => {{1,2,3,4,5,7},{6}}
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => {{1,6},{2,3,4},{5}}
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => {{1,3,5},{2},{4}}
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => {{1,3,4,5,6,7},{2}}
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => {{1,7},{2,3,4,5},{6}}
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => {{1,2,6},{3,4},{5}}
 => 3
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}}
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}}
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
 => 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => {{1,7},{2},{3,4,5,6}}
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => {{1,2,7},{3,4,5},{6}}
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => {{1,5,7},{2,3,4},{6}}
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}}
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}}
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}}
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}}
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => {{1,3,7},{2},{4,5,6}}
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => {{1,6,7},{2},{3,4,5}}
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => {{1,2,3,7},{4,5},{6}}
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => {{1,7},{2,5},{3,4},{6}}
 => 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => {{1,4,5,7},{2,3},{6}}
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => {{1,2,3,4,6,7},{5}}
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}}
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}}
 => 3
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}}
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => {{1,3,4,7},{2},{5,6}}
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
 => 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
 => 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
 => 2
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => {{1,7},{2},{3,6},{4,5}}
 => 2
Description
The number of blocks with odd minimum. See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St001115
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1,4,2] => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,4,2,3] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => 1 = 2 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 0 = 1 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,2,5,4] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => 1 = 2 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,1,3,2] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => 1 = 2 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,1,3,4] => 1 = 2 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0 = 1 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,2,4,6,5] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,4,2,5,3] => 2 = 3 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [4,1,2,5,3] => 1 = 2 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,5,3,4] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,1,3,2,4] => 0 = 1 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,5,2,3,4] => 1 = 2 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,2,4] => 1 = 2 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [6,2,1,3,4,5] => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [1,4,2,3,6,5] => 1 = 2 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [4,1,2,3,6,5] => 0 = 1 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,1,5,2] => 2 = 3 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,1,2,4,3] => 1 = 2 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,1,5,2,3] => 0 = 1 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,5,3,2,4] => 1 = 2 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,1,4] => 1 = 2 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [6,1,3,2,4,5] => 0 = 1 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,1,2,3] => 1 = 2 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [6,3,1,2,4,5] => 1 = 2 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,2,5,3,6,4] => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,3,1,2,6,5] => 1 = 2 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [5,1,2,3,6,4] => 0 = 1 - 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [1,2,3,6,4,5] => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1,5,3,2] => 1 = 2 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => 2 = 3 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,1,4,2] => 2 = 3 - 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [6,1,2,4,3,5] => 1 = 2 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,5,4,2,3] => 1 = 2 - 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,1,4,2,3] => 0 = 1 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,2,1,3] => 1 = 2 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [6,3,2,1,4,5] => 1 = 2 - 1
Description
The number of even descents of a permutation.
Matching statistic: St000257
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 1 = 2 - 1
[1,1]
 => [2]
 => 0 = 1 - 1
[3]
 => [3]
 => 0 = 1 - 1
[2,1]
 => [1,1,1]
 => 1 = 2 - 1
[1,1,1]
 => [2,1]
 => 0 = 1 - 1
[4]
 => [2,2]
 => 1 = 2 - 1
[3,1]
 => [3,1]
 => 0 = 1 - 1
[2,2]
 => [1,1,1,1]
 => 1 = 2 - 1
[2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => 0 = 1 - 1
[5]
 => [5]
 => 0 = 1 - 1
[4,1]
 => [2,2,1]
 => 1 = 2 - 1
[3,2]
 => [3,1,1]
 => 1 = 2 - 1
[3,1,1]
 => [3,2]
 => 0 = 1 - 1
[2,2,1]
 => [1,1,1,1,1]
 => 1 = 2 - 1
[2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [4,1]
 => 0 = 1 - 1
[5,1]
 => [5,1]
 => 0 = 1 - 1
[4,2]
 => [2,2,1,1]
 => 2 = 3 - 1
[4,1,1]
 => [2,2,2]
 => 1 = 2 - 1
[3,3]
 => [6]
 => 0 = 1 - 1
[3,2,1]
 => [3,1,1,1]
 => 1 = 2 - 1
[3,1,1,1]
 => [3,2,1]
 => 0 = 1 - 1
[2,2,2]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
[2,2,1,1]
 => [2,1,1,1,1]
 => 1 = 2 - 1
[2,1,1,1,1]
 => [4,1,1]
 => 1 = 2 - 1
[5,2]
 => [5,1,1]
 => 1 = 2 - 1
[5,1,1]
 => [5,2]
 => 0 = 1 - 1
[4,3]
 => [3,2,2]
 => 1 = 2 - 1
[4,2,1]
 => [2,2,1,1,1]
 => 2 = 3 - 1
[4,1,1,1]
 => [2,2,2,1]
 => 1 = 2 - 1
[3,3,1]
 => [6,1]
 => 0 = 1 - 1
[3,2,2]
 => [3,1,1,1,1]
 => 1 = 2 - 1
[3,2,1,1]
 => [3,2,1,1]
 => 1 = 2 - 1
[3,1,1,1,1]
 => [4,3]
 => 0 = 1 - 1
[2,2,2,1]
 => [1,1,1,1,1,1,1]
 => 1 = 2 - 1
[2,2,1,1,1]
 => [2,1,1,1,1,1]
 => 1 = 2 - 1
[5,3]
 => [5,3]
 => 0 = 1 - 1
[5,2,1]
 => [5,1,1,1]
 => 1 = 2 - 1
[5,1,1,1]
 => [5,2,1]
 => 0 = 1 - 1
[4,4]
 => [2,2,2,2]
 => 1 = 2 - 1
[4,3,1]
 => [3,2,2,1]
 => 1 = 2 - 1
[4,2,2]
 => [2,2,1,1,1,1]
 => 2 = 3 - 1
[4,2,1,1]
 => [2,2,2,1,1]
 => 2 = 3 - 1
[4,1,1,1,1]
 => [4,2,2]
 => 1 = 2 - 1
[3,3,2]
 => [6,1,1]
 => 1 = 2 - 1
[3,3,1,1]
 => [6,2]
 => 0 = 1 - 1
[3,2,2,1]
 => [3,1,1,1,1,1]
 => 1 = 2 - 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => 1 = 2 - 1
[]
 => ?
 => ? = 1 - 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St001114
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [1,2] => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 0 = 1 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [2,1,3] => 1 = 2 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,2,3] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [2,4,5,1,3] => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [4,2,1,3] => 1 = 2 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,3,1,4] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1,2,4] => 1 = 2 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [2,4,5,6,1,3] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,2,5,1,3] => 2 = 3 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,5,1,4] => 1 = 2 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [4,5,1,2,3] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [3,2,1,4] => 1 = 2 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,1,4,5] => 0 = 1 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,2,3,4] => 1 = 2 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,2,4,5] => 1 = 2 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [2,1,3,4,5,6] => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [4,2,5,6,1,3] => 1 = 2 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [2,3,5,6,1,4] => 0 = 1 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,5,2,1,3] => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,5,1,4] => 2 = 3 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,4,1,5] => 1 = 2 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,5,1,2,4] => 0 = 1 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,2,1,3,4] => 1 = 2 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,4,5] => 1 = 2 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [2,3,1,4,5,6] => 0 = 1 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,1,2,3,5] => 1 = 2 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,2,4,5,6] => 1 = 2 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,2,6,1,3] => 0 = 1 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [3,2,5,6,1,4] => 1 = 2 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [2,3,4,6,1,5] => 0 = 1 - 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [4,5,6,1,2,3] => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [3,5,2,1,4] => 1 = 2 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [5,2,3,1,4] => 2 = 3 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,4,1,5] => 2 = 3 - 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [2,3,4,1,5,6] => 1 = 2 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [5,3,1,2,4] => 1 = 2 - 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,4,1,2,5] => 0 = 1 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,2,1,3,5] => 1 = 2 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [3,2,1,4,5,6] => 1 = 2 - 1
[]
 => []
 => [] => [] => ? = 1 - 1
Description
The number of odd descents of a permutation.
Matching statistic: St000386
(load all 2 compositions to match this statistic)
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[3,3]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,2,2]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
[3,3,1]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[2,2,2,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[2,2,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,3,2]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[3,2,2,1]
 => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,4,2,1]
 => [2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 1
[4,3,2,2]
 => [3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 3 - 1
[4,2,2,2,1]
 => [2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 3 - 1
[5,2,2,2,1]
 => [5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 1
[4,3,2,2,1]
 => [3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 3 - 1
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 3 - 1
[4,4,3,2]
 => [3,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 3 - 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 1
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ?
 => ? = 3 - 1
[4,4,3,2,1]
 => [3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ?
 => ? = 3 - 1
[5,4,3,2,1]
 => [5,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => ? = 3 - 1
[]
 => ?
 => ?
 => ?
 => ? = 1 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000201
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [[.,.],.]
 => 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 1
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => 1
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 2
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 2
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => 1
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 3
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2
[3,3]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 1
[2,2,2]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => 2
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => 2
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => 2
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => 1
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => 3
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2
[3,3,1]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => 2
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[2,2,2,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => 2
[2,2,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,.],.]]]]]
 => 2
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => 1
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],[.,[.,.]]]],.]
 => 2
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => 1
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => 2
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2
[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],[.,.]]]]]
 => 3
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => 3
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[3,3,2]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],[.,.]]]]],.]
 => 2
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,[.,.]],.]]]],.]
 => 1
[3,2,2,1]
 => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,[.,.]],.]]]]
 => 2
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => 2
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
 => ? = 2
[4,4,2,1]
 => [2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => ? = 3
[4,3,2,2]
 => [3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => ? = 3
[4,2,2,2,1]
 => [2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
 => ? = 3
[5,2,2,2,1]
 => [5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
 => ? = 2
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
 => ? = 3
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => ? = 3
[4,3,2,2,1]
 => [3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,.],[[.,.],.]]]]]]
 => ? = 3
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => ? = 3
[5,3,2,2,1]
 => [5,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
 => ? = 2
[4,4,3,2]
 => [3,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
 => ? = 3
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]
 => ? = 2
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
 => ? = 3
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ?
 => ? = 3
[4,4,3,2,1]
 => [3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,[.,[[.,.],.]]]]]]
 => ? = 3
[5,4,3,2,1]
 => [5,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => ? = 3
[]
 => ?
 => ?
 => ?
 => ? = 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000196
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [[.,.],.]
 => 0 = 1 - 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 1 = 2 - 1
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 0 = 1 - 1
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 0 = 1 - 1
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0 = 1 - 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 1 = 2 - 1
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 0 = 1 - 1
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 1 = 2 - 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 0 = 1 - 1
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => 0 = 1 - 1
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 1 = 2 - 1
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 1 = 2 - 1
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 0 = 1 - 1
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 1 = 2 - 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => 0 = 1 - 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => 0 = 1 - 1
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2 = 3 - 1
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 1 = 2 - 1
[3,3]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => 0 = 1 - 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 1 = 2 - 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 0 = 1 - 1
[2,2,2]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => 1 = 2 - 1
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => 1 = 2 - 1
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 1 = 2 - 1
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => 1 = 2 - 1
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => 0 = 1 - 1
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 1 = 2 - 1
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => 2 = 3 - 1
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 1 = 2 - 1
[3,3,1]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[[.,.],.]]]]],.]
 => 0 = 1 - 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => 1 = 2 - 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 1 = 2 - 1
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 0 = 1 - 1
[2,2,2,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ? = 2 - 1
[2,2,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,.],.]]]]]
 => 1 = 2 - 1
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => 0 = 1 - 1
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],[.,[.,.]]]],.]
 => 1 = 2 - 1
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => 0 = 1 - 1
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => 1 = 2 - 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 1 = 2 - 1
[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],[.,.]]]]]
 => 2 = 3 - 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => 2 = 3 - 1
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 1 = 2 - 1
[3,3,2]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],[.,.]]]]],.]
 => 1 = 2 - 1
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,[.,.]],.]]]],.]
 => 0 = 1 - 1
[3,2,2,1]
 => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,[.,.]],.]]]]
 => 1 = 2 - 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => 1 = 2 - 1
[2,2,2,2]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 2 - 1
[2,2,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
 => ? = 2 - 1
[5,4]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[.,[[.,[.,.]],[.,.]]],.]
 => 1 = 2 - 1
[4,2,2,1]
 => [2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
 => ? = 3 - 1
[3,2,2,2]
 => [3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
 => ? = 2 - 1
[2,2,2,2,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => ? = 2 - 1
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
 => ? = 3 - 1
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
 => ? = 3 - 1
[3,2,2,2,1]
 => [3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
 => ? = 2 - 1
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
 => ? = 2 - 1
[4,4,2,1]
 => [2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => ? = 3 - 1
[4,3,2,2]
 => [3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => ? = 3 - 1
[4,2,2,2,1]
 => [2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
 => ? = 3 - 1
[5,2,2,2,1]
 => [5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[.,.],[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
 => ? = 2 - 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
 => ? = 3 - 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => ? = 3 - 1
[4,3,2,2,1]
 => [3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,.],[[.,.],.]]]]]]
 => ? = 3 - 1
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => ? = 3 - 1
[5,3,2,2,1]
 => [5,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
 => ? = 2 - 1
[4,4,3,2]
 => [3,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
 => ? = 3 - 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]
 => ? = 2 - 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
 => ? = 3 - 1
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ?
 => ? = 3 - 1
[4,4,3,2,1]
 => [3,2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => [[.,.],[.,[[.,.],[.,[.,[[.,.],.]]]]]]
 => ? = 3 - 1
[5,4,3,2,1]
 => [5,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [[.,.],[.,[[[.,.],[[.,.],.]],.]]]
 => ? = 3 - 1
[]
 => ?
 => ?
 => ?
 => ? = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000256
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[1,1]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[3]
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[2,1]
 => [1,1,1]
 => [3]
 => 1 = 2 - 1
[1,1,1]
 => [2,1]
 => [2,1]
 => 0 = 1 - 1
[4]
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[3,1]
 => [3,1]
 => [2,1,1]
 => 0 = 1 - 1
[2,2]
 => [1,1,1,1]
 => [4]
 => 1 = 2 - 1
[2,1,1]
 => [2,1,1]
 => [3,1]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[5]
 => [5]
 => [1,1,1,1,1]
 => 0 = 1 - 1
[4,1]
 => [2,2,1]
 => [3,2]
 => 1 = 2 - 1
[3,2]
 => [3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
[3,1,1]
 => [3,2]
 => [2,2,1]
 => 0 = 1 - 1
[2,2,1]
 => [1,1,1,1,1]
 => [5]
 => 1 = 2 - 1
[2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 0 = 1 - 1
[5,1]
 => [5,1]
 => [2,1,1,1,1]
 => 0 = 1 - 1
[4,2]
 => [2,2,1,1]
 => [4,2]
 => 2 = 3 - 1
[4,1,1]
 => [2,2,2]
 => [3,3]
 => 1 = 2 - 1
[3,3]
 => [6]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
[3,2,1]
 => [3,1,1,1]
 => [4,1,1]
 => 1 = 2 - 1
[3,1,1,1]
 => [3,2,1]
 => [3,2,1]
 => 0 = 1 - 1
[2,2,2]
 => [1,1,1,1,1,1]
 => [6]
 => 1 = 2 - 1
[2,2,1,1]
 => [2,1,1,1,1]
 => [5,1]
 => 1 = 2 - 1
[2,1,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => 1 = 2 - 1
[5,2]
 => [5,1,1]
 => [3,1,1,1,1]
 => 1 = 2 - 1
[5,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => 0 = 1 - 1
[4,3]
 => [3,2,2]
 => [3,3,1]
 => 1 = 2 - 1
[4,2,1]
 => [2,2,1,1,1]
 => [5,2]
 => 2 = 3 - 1
[4,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => 1 = 2 - 1
[3,3,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => 0 = 1 - 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => 1 = 2 - 1
[3,2,1,1]
 => [3,2,1,1]
 => [4,2,1]
 => 1 = 2 - 1
[3,1,1,1,1]
 => [4,3]
 => [2,2,2,1]
 => 0 = 1 - 1
[2,2,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => 1 = 2 - 1
[2,2,1,1,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => 1 = 2 - 1
[5,3]
 => [5,3]
 => [2,2,2,1,1]
 => 0 = 1 - 1
[5,2,1]
 => [5,1,1,1]
 => [4,1,1,1,1]
 => 1 = 2 - 1
[5,1,1,1]
 => [5,2,1]
 => [3,2,1,1,1]
 => 0 = 1 - 1
[4,4]
 => [2,2,2,2]
 => [4,4]
 => 1 = 2 - 1
[4,3,1]
 => [3,2,2,1]
 => [4,3,1]
 => 1 = 2 - 1
[4,2,2]
 => [2,2,1,1,1,1]
 => [6,2]
 => 2 = 3 - 1
[4,2,1,1]
 => [2,2,2,1,1]
 => [5,3]
 => 2 = 3 - 1
[4,1,1,1,1]
 => [4,2,2]
 => [3,3,1,1]
 => 1 = 2 - 1
[3,3,2]
 => [6,1,1]
 => [3,1,1,1,1,1]
 => 1 = 2 - 1
[3,3,1,1]
 => [6,2]
 => [2,2,1,1,1,1]
 => 0 = 1 - 1
[3,2,2,1]
 => [3,1,1,1,1,1]
 => [6,1,1]
 => 1 = 2 - 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [5,2,1]
 => 1 = 2 - 1
[5,4,2]
 => [5,2,2,1,1]
 => [5,3,1,1,1]
 => ? = 3 - 1
[5,4,1,1]
 => [5,2,2,2]
 => [4,4,1,1,1]
 => ? = 2 - 1
[5,3,3]
 => [6,5]
 => [2,2,2,2,2,1]
 => ? = 1 - 1
[5,3,2,1]
 => [5,3,1,1,1]
 => [5,2,2,1,1]
 => ? = 2 - 1
[5,3,1,1,1]
 => [5,3,2,1]
 => [4,3,2,1,1]
 => ? = 1 - 1
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [7,1,1,1,1]
 => ? = 2 - 1
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [6,2,1,1,1]
 => ? = 2 - 1
[4,4,3]
 => [3,2,2,2,2]
 => [5,5,1]
 => ? = 2 - 1
[4,4,2,1]
 => [2,2,2,2,1,1,1]
 => [7,4]
 => ? = 3 - 1
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [6,5]
 => ? = 2 - 1
[4,3,3,1]
 => [6,2,2,1]
 => [4,3,1,1,1,1]
 => ? = 2 - 1
[4,3,2,2]
 => [3,2,2,1,1,1,1]
 => [7,3,1]
 => ? = 3 - 1
[4,3,2,1,1]
 => [3,2,2,2,1,1]
 => [6,4,1]
 => ? = 3 - 1
[4,2,2,2,1]
 => [2,2,1,1,1,1,1,1,1]
 => [9,2]
 => ? = 3 - 1
[3,3,3,2]
 => [6,3,1,1]
 => [4,2,2,1,1,1]
 => ? = 2 - 1
[3,3,3,1,1]
 => [6,3,2]
 => [3,3,2,1,1,1]
 => ? = 1 - 1
[3,3,2,2,1]
 => [6,1,1,1,1,1]
 => [6,1,1,1,1,1]
 => ? = 2 - 1
[5,4,3]
 => [5,3,2,2]
 => [4,4,2,1,1]
 => ? = 2 - 1
[5,4,2,1]
 => [5,2,2,1,1,1]
 => [6,3,1,1,1]
 => ? = 3 - 1
[5,4,1,1,1]
 => [5,2,2,2,1]
 => [5,4,1,1,1]
 => ? = 2 - 1
[5,3,3,1]
 => [6,5,1]
 => [3,2,2,2,2,1]
 => ? = 1 - 1
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [6,2,2,1,1]
 => ? = 2 - 1
[5,3,2,1,1]
 => [5,3,2,1,1]
 => [5,3,2,1,1]
 => ? = 2 - 1
[5,2,2,2,1]
 => [5,1,1,1,1,1,1,1]
 => [8,1,1,1,1]
 => ? = 2 - 1
[4,4,3,1]
 => [3,2,2,2,2,1]
 => [6,5,1]
 => ? = 2 - 1
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [8,4]
 => ? = 3 - 1
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [7,5]
 => ? = 3 - 1
[4,3,3,2]
 => [6,2,2,1,1]
 => [5,3,1,1,1,1]
 => ? = 3 - 1
[4,3,3,1,1]
 => [6,2,2,2]
 => [4,4,1,1,1,1]
 => ? = 2 - 1
[4,3,2,2,1]
 => [3,2,2,1,1,1,1,1]
 => [8,3,1]
 => ? = 3 - 1
[3,3,3,2,1]
 => [6,3,1,1,1]
 => [5,2,2,1,1,1]
 => ? = 2 - 1
[5,4,3,1]
 => [5,3,2,2,1]
 => [5,4,2,1,1]
 => ? = 2 - 1
[5,4,2,2]
 => [5,2,2,1,1,1,1]
 => [7,3,1,1,1]
 => ? = 3 - 1
[5,4,2,1,1]
 => [5,2,2,2,1,1]
 => [6,4,1,1,1]
 => ? = 3 - 1
[5,3,3,2]
 => [6,5,1,1]
 => [4,2,2,2,2,1]
 => ? = 2 - 1
[5,3,3,1,1]
 => [6,5,2]
 => [3,3,2,2,2,1]
 => ? = 1 - 1
[5,3,2,2,1]
 => [5,3,1,1,1,1,1]
 => [7,2,2,1,1]
 => ? = 2 - 1
[4,4,3,2]
 => [3,2,2,2,2,1,1]
 => [7,5,1]
 => ? = 3 - 1
[4,4,3,1,1]
 => [3,2,2,2,2,2]
 => [6,6,1]
 => ? = 2 - 1
[4,4,2,2,1]
 => [2,2,2,2,1,1,1,1,1]
 => [9,4]
 => ? = 3 - 1
[4,3,3,2,1]
 => [6,2,2,1,1,1]
 => [6,3,1,1,1,1]
 => ? = 3 - 1
[5,4,3,2]
 => [5,3,2,2,1,1]
 => [6,4,2,1,1]
 => ? = 3 - 1
[5,4,3,1,1]
 => [5,3,2,2,2]
 => [5,5,2,1,1]
 => ? = 2 - 1
[5,4,2,2,1]
 => [5,2,2,1,1,1,1,1]
 => [8,3,1,1,1]
 => ? = 3 - 1
[5,3,3,2,1]
 => [6,5,1,1,1]
 => [5,2,2,2,2,1]
 => ? = 2 - 1
[4,4,3,2,1]
 => [3,2,2,2,2,1,1,1]
 => [8,5,1]
 => ? = 3 - 1
[5,4,3,2,1]
 => [5,3,2,2,1,1,1]
 => [7,4,2,1,1]
 => ? = 3 - 1
[]
 => ?
 => ?
 => ? = 1 - 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000099
Mapping
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1
[3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[2,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 2
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[2,2]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 2
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[5]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 1
[4,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
[3,2]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[3,1,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[2,2,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 2
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,1,1,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 1
[5,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 1
[4,2]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 3
[4,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2
[3,3]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 2
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
[2,2,2]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 2
[2,2,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 2
[2,1,1,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 2
[5,2]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 2
[5,1,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 1
[4,3]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 2
[4,2,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 3
[4,1,1,1]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 2
[3,3,1]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ? = 1
[3,2,2]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 2
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 2
[3,1,1,1,1]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 1
[2,2,2,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ? = 2
[2,2,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ? = 2
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 1
[5,2,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 2
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 1
[4,4]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => 2
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 2
[4,2,2]
 => [2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => ? = 3
[4,2,1,1]
 => [2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => 3
[4,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
[3,3,2]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ? = 2
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 1
[3,2,2,1]
 => [3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => ? = 2
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => 2
[2,2,2,2]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => ? = 2
[2,2,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => ? = 2
[5,4]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => 2
[5,3,1]
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [6,4,2,1,3,5] => 1
[5,2,2]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => 2
[5,2,1,1]
 => [5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,1,5] => 2
[5,1,1,1,1]
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,2,3,4] => 1
[4,4,1]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => 2
[4,3,2]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => 3
[4,3,1,1]
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,1,2] => 2
[4,2,2,1]
 => [2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => ? = 3
[4,2,1,1,1]
 => [2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => ? = 3
[3,3,3]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => ? = 1
[3,3,2,1]
 => [6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,1,5,6] => ? = 2
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => ? = 1
[3,2,2,2]
 => [3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => ? = 2
[3,2,2,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,7,1] => ? = 2
[2,2,2,2,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => ? = 2
[5,4,1]
 => [5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,1,5] => 2
[5,2,2,1]
 => [5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [6,2,3,4,5,7,1] => ? = 2
[4,4,2]
 => [2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,7,1] => ? = 3
[4,4,1,1]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => ? = 2
[4,3,3]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [7,3,4,1,2,5,6] => ? = 2
[4,3,2,1]
 => [3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,7,1] => ? = 3
[4,2,2,2]
 => [2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => ? = 3
[4,2,2,1,1]
 => [2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,1] => ? = 3
[3,3,3,1]
 => [6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => ? = 1
[3,3,2,2]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,1,6] => ? = 2
[3,3,2,1,1]
 => [6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,1,5,6] => ? = 2
[3,2,2,2,1]
 => [3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => ? = 2
[5,3,3]
 => [6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [7,6,1,2,3,4,5] => ? = 1
[5,2,2,2]
 => [5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => ? = 2
[5,2,2,1,1]
 => [5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => ? = 2
[4,4,3]
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,1,2] => ? = 2
[4,4,2,1]
 => [2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ? = 3
[4,4,1,1,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 2
[4,3,3,1]
 => [6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [7,3,4,2,1,5,6] => ? = 2
[4,3,2,2]
 => [3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,3,5,2,6,7,8,1] => ? = 3
[4,3,2,1,1]
 => [3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,7,1] => ? = 3
[4,2,2,2,1]
 => [2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ? = 3
[3,3,3,2]
 => [6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [7,4,2,3,1,5,6] => ? = 2
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => ? = 1
[3,3,2,2,1]
 => [6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => ? = 2
[5,4,2,1]
 => [5,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [6,3,4,2,5,7,1] => ? = 3
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [7,6,2,1,3,4,5] => ? = 1
[5,3,2,2]
 => [5,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [6,4,2,3,5,7,1] => ? = 2
[5,2,2,2,1]
 => [5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ? = 2
[4,4,3,1]
 => [3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => ? = 2
[4,4,2,2]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ? = 3
[4,4,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,2,8,1] => ? = 3
Description
The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000659The number of rises of length at least 2 of a Dyck path. St000023The number of inner peaks of a permutation. St000779The tier of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001729The number of visible descents of a permutation.