Your data matches 53 different statistics following compositions of up to 3 maps.
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Matching statistic: St000790
(load all 3 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000790: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
 => 0
[2] => [1,1,0,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => 0
[1,2] => [1,0,1,1,0,0]
 => 0
[2,1] => [1,1,0,0,1,0]
 => 0
[3] => [1,1,1,0,0,0]
 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 0
Description
The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. Apparently, the total number of these is given in [1]. The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St000573
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000573: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
 => [2,1] => {{1,2}}
 => 0
[2] => [1,1,0,0]
 => [1,2] => {{1},{2}}
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [2,1,3] => {{1,2},{3}}
 => 0
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 0
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => {{1,2,3}}
 => 0
[3] => [1,1,1,0,0,0]
 => [1,2,3] => {{1},{2},{3}}
 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => {{1,2,3,4}}
 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => {{1,2,3,4}}
 => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => {{1,2,3,4}}
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => {{1,2,3,4},{5}}
 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => {{1,2},{3,4,5}}
 => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => {{1,2,3,4,5}}
 => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => {{1,2,3,4},{5}}
 => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => {{1,3},{2,4},{5}}
 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => {{1,2,3,4,5}}
 => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => {{1,2,3,4,5}}
 => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => {{1,2,4},{3,5}}
 => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => {{1,2,3,4,5}}
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => {{1,2,3,4,5}}
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}}
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5] => {{1,2,3,4},{5,6}}
 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,6,3,5] => {{1,2},{3,4,5,6}}
 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => {{1,2,4},{3,5,6}}
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => {{1,2},{3,4},{5},{6}}
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,4,1,3,5,6] => {{1,2,3,4},{5},{6}}
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,4,5,6,3] => {{1,2},{3,4,5,6}}
 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
 => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,5,3,6,4] => {{1,2},{3,4,5,6}}
 => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => {{1,2,3,4,5,6}}
 => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,5,6,3,4] => {{1,2},{3,5},{4,6}}
 => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,5,6,1,3,4] => {{1,2,3,4,5,6}}
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,6,3,4,5] => {{1,2},{3,4,5,6}}
 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,6,1,3,4,5] => {{1,2,3,4,5,6}}
 => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,5,6] => {{1,2},{3},{4},{5},{6}}
 => 6
[1,5] => [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
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,4,2,6,5] => {{1,2,3,4},{5,6}}
 => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [3,4,1,2,6,5] => {{1,3},{2,4},{5,6}}
 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [3,1,4,6,2,5] => {{1,2,3,4,5,6}}
 => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,4,6,1,2,5] => {{1,2,3,4,5,6}}
 => 0
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. This is the number of pairs $i\lt j$ in different blocks such that $i$ is a singleton block and $j$ is the maximal element of a block.
Matching statistic: St000575
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000575: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
 => [2,1] => {{1,2}}
 => 0
[2] => [1,1,0,0]
 => [1,2] => {{1},{2}}
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => {{1,3},{2}}
 => 0
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 0
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => {{1,2,3}}
 => 0
[3] => [1,1,1,0,0,0]
 => [1,2,3] => {{1},{2},{3}}
 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => {{1,4},{2,3}}
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => {{1,2,3,4}}
 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => {{1,4},{2},{3}}
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => {{1,2,3,4}}
 => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => {{1,2,3,4}}
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => {{1,2,4,5},{3}}
 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => {{1,5},{2,3,4}}
 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => {{1,3,5},{2,4}}
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => {{1,5},{2,3,4}}
 => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => {{1,2,3,4,5}}
 => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => {{1,2,4,5},{3}}
 => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => {{1,4},{2,5},{3}}
 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => {{1,2,3,4,5}}
 => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => {{1,2,3,4,5}}
 => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => {{1,3,5},{2,4}}
 => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => {{1,2,3,4,5}}
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => {{1,2,3,4,5}}
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}}
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => {{1,2,5,6},{3,4}}
 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => {{1,6},{2,3,4,5}}
 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => {{1,3,4,6},{2,5}}
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => {{1,6},{2,5},{3},{4}}
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => {{1,2,5,6},{3},{4}}
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => {{1,6},{2,3,4,5}}
 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => {{1,2,3,4,5,6}}
 => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => {{1,6},{2,3,4,5}}
 => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => {{1,2,3,4,5,6}}
 => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => {{1,6},{2,4},{3,5}}
 => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => {{1,2,3,4,5,6}}
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => {{1,6},{2,3,4,5}}
 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => {{1,2,3,4,5,6}}
 => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => {{1,6},{2},{3},{4},{5}}
 => 6
[1,5] => [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
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => {{1,2,5,6},{3,4}}
 => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => {{1,5},{2,6},{3,4}}
 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => {{1,2,3,4,5,6}}
 => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => {{1,2,3,4,5,6}}
 => 0
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block and $j$ is a singleton block.
Matching statistic: St000576
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000576: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
 => [2,1] => {{1,2}}
 => 0
[2] => [1,1,0,0]
 => [1,2] => {{1},{2}}
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => {{1,3},{2}}
 => 0
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 0
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => {{1,2,3}}
 => 0
[3] => [1,1,1,0,0,0]
 => [1,2,3] => {{1},{2},{3}}
 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => {{1,4},{2,3}}
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => {{1,2,3,4}}
 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => {{1,4},{2},{3}}
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => {{1,2,3,4}}
 => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => {{1,2,3,4}}
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => {{1,2,4,5},{3}}
 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => {{1,5},{2,3,4}}
 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => {{1,3,5},{2,4}}
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => {{1,5},{2,3,4}}
 => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => {{1,2,3,4,5}}
 => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => {{1,2,4,5},{3}}
 => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => {{1,4},{2,5},{3}}
 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => {{1,2,3,4,5}}
 => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => {{1,2,3,4,5}}
 => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => {{1,3,5},{2,4}}
 => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => {{1,2,3,4,5}}
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => {{1,2,3,4,5}}
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}}
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => {{1,2,5,6},{3,4}}
 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => {{1,6},{2,3,4,5}}
 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => {{1,3,4,6},{2,5}}
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => {{1,6},{2,5},{3},{4}}
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => {{1,2,5,6},{3},{4}}
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => {{1,6},{2,3,4,5}}
 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => {{1,2,3,4,5,6}}
 => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => {{1,6},{2,3,4,5}}
 => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => {{1,2,3,4,5,6}}
 => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => {{1,6},{2,4},{3,5}}
 => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => {{1,2,3,4,5,6}}
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => {{1,6},{2,3,4,5}}
 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => {{1,2,3,4,5,6}}
 => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => {{1,6},{2},{3},{4},{5}}
 => 6
[1,5] => [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
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => {{1,2,5,6},{3,4}}
 => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => {{1,5},{2,6},{3,4}}
 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => {{1,2,3,4,5,6}}
 => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => {{1,2,3,4,5,6}}
 => 0
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block and $j$ is the minimal element of a block.
Matching statistic: St000578
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000578: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
 => [2,1] => {{1,2}}
 => 0
[2] => [1,1,0,0]
 => [1,2] => {{1},{2}}
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [2,1,3] => {{1,2},{3}}
 => 0
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 0
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => {{1,2,3}}
 => 0
[3] => [1,1,1,0,0,0]
 => [1,2,3] => {{1},{2},{3}}
 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => {{1,2,3,4}}
 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => {{1,2,3,4}}
 => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => {{1,2,3,4}}
 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => {{1,2,3,4},{5}}
 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => {{1,2},{3,4,5}}
 => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => {{1,2,3,4,5}}
 => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => {{1,2,3,4},{5}}
 => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => {{1,3},{2,4},{5}}
 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => {{1,2,3,4,5}}
 => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => {{1,2,3,4,5}}
 => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => {{1,2,4},{3,5}}
 => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => {{1,2,3,4,5}}
 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => {{1,2,3,4,5}}
 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}}
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5] => {{1,2,3,4},{5,6}}
 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,6,3,5] => {{1,2},{3,4,5,6}}
 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => {{1,2,4},{3,5,6}}
 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => {{1,2},{3,4},{5},{6}}
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,4,1,3,5,6] => {{1,2,3,4},{5},{6}}
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,4,5,6,3] => {{1,2},{3,4,5,6}}
 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3] => {{1,2,3,4,5,6}}
 => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,5,3,6,4] => {{1,2},{3,4,5,6}}
 => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => {{1,2,3,4,5,6}}
 => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,5,6,3,4] => {{1,2},{3,5},{4,6}}
 => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,5,6,1,3,4] => {{1,2,3,4,5,6}}
 => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,6,3,4,5] => {{1,2},{3,4,5,6}}
 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,6,1,3,4,5] => {{1,2,3,4,5,6}}
 => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,5,6] => {{1,2},{3},{4},{5},{6}}
 => 6
[1,5] => [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
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,4,2,6,5] => {{1,2,3,4},{5,6}}
 => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [3,4,1,2,6,5] => {{1,3},{2,4},{5,6}}
 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [3,1,4,6,2,5] => {{1,2,3,4,5,6}}
 => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,4,6,1,2,5] => {{1,2,3,4,5,6}}
 => 0
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. This is the number of pairs $i\lt j$ such that $i$ is a singleton block.
Matching statistic: St000205
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 79%●distinct values known / distinct values provided: 14%
Values
[1,1] => [1,1]
 => [1]
 => []
 => ? ∊ {0,1}
[2] => [2]
 => []
 => ?
 => ? ∊ {0,1}
[1,1,1] => [1,1,1]
 => [1,1]
 => [1]
 => 0
[1,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,3}
[2,1] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,3}
[3] => [3]
 => []
 => ?
 => ? ∊ {0,0,3}
[1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => 0
[1,2,1] => [2,1,1]
 => [1,1]
 => [1]
 => 0
[1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,1,6}
[2,1,1] => [2,1,1]
 => [1,1]
 => [1]
 => 0
[2,2] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,1,6}
[3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,1,6}
[4] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,6}
[1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,3] => [3,1,1]
 => [1,1]
 => [1]
 => 0
[1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,2] => [2,2,1]
 => [2,1]
 => [1]
 => 0
[1,3,1] => [3,1,1]
 => [1,1]
 => [1]
 => 0
[1,4] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,3,10}
[2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,2] => [2,2,1]
 => [2,1]
 => [1]
 => 0
[2,2,1] => [2,2,1]
 => [2,1]
 => [1]
 => 0
[2,3] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,3,10}
[3,1,1] => [3,1,1]
 => [1,1]
 => [1]
 => 0
[3,2] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,3,10}
[4,1] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,3,10}
[5] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,3,10}
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,4] => [4,1,1]
 => [1,1]
 => [1]
 => 0
[1,2,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,3] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[1,3,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,2] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[1,4,1] => [4,1,1]
 => [1,1]
 => [1]
 => 0
[1,5] => [5,1]
 => [1]
 => []
 => ? ∊ {1,1,1,1,6,15}
[2,1,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,3] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[2,2,1,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,2,2] => [2,2,2]
 => [2,2]
 => [2]
 => 0
[2,3,1] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[2,4] => [4,2]
 => [2]
 => []
 => ? ∊ {1,1,1,1,6,15}
[3,1,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,1,2] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[3,2,1] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[3,3] => [3,3]
 => [3]
 => []
 => ? ∊ {1,1,1,1,6,15}
[4,1,1] => [4,1,1]
 => [1,1]
 => [1]
 => 0
[4,2] => [4,2]
 => [2]
 => []
 => ? ∊ {1,1,1,1,6,15}
[5,1] => [5,1]
 => [1]
 => []
 => ? ∊ {1,1,1,1,6,15}
[6] => [6]
 => []
 => ?
 => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,1,1,1,1,2] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,2,1] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,2,1,1] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,2,2] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,3,1] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,4] => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,6] => [6,1]
 => [1]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[2,5] => [5,2]
 => [2]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[3,4] => [4,3]
 => [3]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[4,3] => [4,3]
 => [3]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[5,2] => [5,2]
 => [2]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[6,1] => [6,1]
 => [1]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[7] => [7]
 => []
 => ?
 => ? ∊ {0,3,3,3,3,10,21}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 79%●distinct values known / distinct values provided: 14%
Values
[1,1] => [1,1]
 => [1]
 => []
 => ? ∊ {0,1}
[2] => [2]
 => []
 => ?
 => ? ∊ {0,1}
[1,1,1] => [1,1,1]
 => [1,1]
 => [1]
 => 0
[1,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,3}
[2,1] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,3}
[3] => [3]
 => []
 => ?
 => ? ∊ {0,0,3}
[1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => 0
[1,2,1] => [2,1,1]
 => [1,1]
 => [1]
 => 0
[1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,1,6}
[2,1,1] => [2,1,1]
 => [1,1]
 => [1]
 => 0
[2,2] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,1,6}
[3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,1,6}
[4] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,6}
[1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,3] => [3,1,1]
 => [1,1]
 => [1]
 => 0
[1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,2,2] => [2,2,1]
 => [2,1]
 => [1]
 => 0
[1,3,1] => [3,1,1]
 => [1,1]
 => [1]
 => 0
[1,4] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,3,10}
[2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,2] => [2,2,1]
 => [2,1]
 => [1]
 => 0
[2,2,1] => [2,2,1]
 => [2,1]
 => [1]
 => 0
[2,3] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,3,10}
[3,1,1] => [3,1,1]
 => [1,1]
 => [1]
 => 0
[3,2] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,3,10}
[4,1] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,3,10}
[5] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,3,10}
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,4] => [4,1,1]
 => [1,1]
 => [1]
 => 0
[1,2,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[1,2,3] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[1,3,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,3,2] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[1,4,1] => [4,1,1]
 => [1,1]
 => [1]
 => 0
[1,5] => [5,1]
 => [1]
 => []
 => ? ∊ {1,1,1,1,6,15}
[2,1,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,3] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[2,2,1,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,2,2] => [2,2,2]
 => [2,2]
 => [2]
 => 0
[2,3,1] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[2,4] => [4,2]
 => [2]
 => []
 => ? ∊ {1,1,1,1,6,15}
[3,1,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,1,2] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[3,2,1] => [3,2,1]
 => [2,1]
 => [1]
 => 0
[3,3] => [3,3]
 => [3]
 => []
 => ? ∊ {1,1,1,1,6,15}
[4,1,1] => [4,1,1]
 => [1,1]
 => [1]
 => 0
[4,2] => [4,2]
 => [2]
 => []
 => ? ∊ {1,1,1,1,6,15}
[5,1] => [5,1]
 => [1]
 => []
 => ? ∊ {1,1,1,1,6,15}
[6] => [6]
 => []
 => ?
 => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,1,1,1,1,2] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,2,1] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,2,1,1] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,2,2] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,3,1] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,4] => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,6] => [6,1]
 => [1]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[2,5] => [5,2]
 => [2]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[3,4] => [4,3]
 => [3]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[4,3] => [4,3]
 => [3]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[5,2] => [5,2]
 => [2]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[6,1] => [6,1]
 => [1]
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[7] => [7]
 => []
 => ?
 => ? ∊ {0,3,3,3,3,10,21}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000296
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 79%●distinct values known / distinct values provided: 14%
Values
[1,1] => [[1,1],[]]
 => []
 => => ? ∊ {0,1}
[2] => [[2],[]]
 => []
 => => ? ∊ {0,1}
[1,1,1] => [[1,1,1],[]]
 => []
 => => ? ∊ {0,0,3}
[1,2] => [[2,1],[]]
 => []
 => => ? ∊ {0,0,3}
[2,1] => [[2,2],[1]]
 => [1]
 => 10 => 0
[3] => [[3],[]]
 => []
 => => ? ∊ {0,0,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => => ? ∊ {0,0,1,6}
[1,1,2] => [[2,1,1],[]]
 => []
 => => ? ∊ {0,0,1,6}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 10 => 0
[1,3] => [[3,1],[]]
 => []
 => => ? ∊ {0,0,1,6}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 110 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => 10 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => 100 => 0
[4] => [[4],[]]
 => []
 => => ? ∊ {0,0,1,6}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,3] => [[3,1,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 10 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 100 => 0
[1,4] => [[4,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 110 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1010 => 0
[2,3] => [[4,2],[1]]
 => [1]
 => 10 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1100 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => 100 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => 1000 => 0
[5] => [[5],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 100 => 0
[1,1,4] => [[4,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1010 => 0
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 10 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1100 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 100 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1000 => 0
[1,5] => [[5,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 11110 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 10110 => 0
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 110 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 11010 => 0
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1010 => 0
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 10010 => 0
[2,4] => [[5,2],[1]]
 => [1]
 => 10 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 11100 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1100 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 10100 => 0
[3,3] => [[5,3],[2]]
 => [2]
 => 100 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 11000 => 0
[4,2] => [[5,4],[3]]
 => [3]
 => 1000 => 0
[5,1] => [[5,5],[4]]
 => [4]
 => 10000 => 0
[6] => [[6],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => 100 => 0
[1,1,1,4] => [[4,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => 1010 => 0
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,5] => [[5,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,6] => [[6,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[7] => [[7],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
Description
The length of the symmetric border of a binary word. The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix. The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 2 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 79%●distinct values known / distinct values provided: 14%
Values
[1,1] => [[1,1],[]]
 => []
 => => ? ∊ {0,1}
[2] => [[2],[]]
 => []
 => => ? ∊ {0,1}
[1,1,1] => [[1,1,1],[]]
 => []
 => => ? ∊ {0,0,3}
[1,2] => [[2,1],[]]
 => []
 => => ? ∊ {0,0,3}
[2,1] => [[2,2],[1]]
 => [1]
 => 10 => 0
[3] => [[3],[]]
 => []
 => => ? ∊ {0,0,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => => ? ∊ {0,0,1,6}
[1,1,2] => [[2,1,1],[]]
 => []
 => => ? ∊ {0,0,1,6}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 10 => 0
[1,3] => [[3,1],[]]
 => []
 => => ? ∊ {0,0,1,6}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 110 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => 10 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => 100 => 0
[4] => [[4],[]]
 => []
 => => ? ∊ {0,0,1,6}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,3] => [[3,1,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 10 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 100 => 0
[1,4] => [[4,1],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 110 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1010 => 0
[2,3] => [[4,2],[1]]
 => [1]
 => 10 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1100 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => 100 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => 1000 => 0
[5] => [[5],[]]
 => []
 => => ? ∊ {0,0,0,3,10}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 100 => 0
[1,1,4] => [[4,1,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1010 => 0
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 10 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1100 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 100 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1000 => 0
[1,5] => [[5,1],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 11110 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 10110 => 0
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 110 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 11010 => 0
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1010 => 0
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 10010 => 0
[2,4] => [[5,2],[1]]
 => [1]
 => 10 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 11100 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1100 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 10100 => 0
[3,3] => [[5,3],[2]]
 => [2]
 => 100 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 11000 => 0
[4,2] => [[5,4],[3]]
 => [3]
 => 1000 => 0
[5,1] => [[5,5],[4]]
 => [4]
 => 10000 => 0
[6] => [[6],[]]
 => []
 => => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => 100 => 0
[1,1,1,4] => [[4,1,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => 1010 => 0
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 10 => 0
[1,1,5] => [[5,1,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[1,6] => [[6,1],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
[7] => [[7],[]]
 => []
 => => ? ∊ {0,3,3,3,3,10,21}
Description
The defect of a binary word. The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St000687
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 79%●distinct values known / distinct values provided: 14%
Values
[1,1] => [[1,1],[]]
 => []
 => []
 => ? ∊ {0,1}
[2] => [[2],[]]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1] => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,3}
[1,2] => [[2,1],[]]
 => []
 => []
 => ? ∊ {0,0,3}
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0]
 => 0
[3] => [[3],[]]
 => []
 => []
 => ? ∊ {0,0,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,6}
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,6}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3] => [[3,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,6}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[4] => [[4],[]]
 => []
 => []
 => ? ∊ {0,0,1,6}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,3,10}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,3,10}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,3,10}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,4] => [[4,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,3,10}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[5] => [[5],[]]
 => []
 => []
 => ? ∊ {0,0,0,3,10}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,6,15}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,6,15}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,6,15}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,5] => [[5,1],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,6,15}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3] => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
[4,2] => [[5,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[5,1] => [[5,5],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[6] => [[6],[]]
 => []
 => []
 => ? ∊ {1,1,1,1,6,15}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,4] => [[4,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,5] => [[5,1,1],[]]
 => []
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[1,6] => [[6,1],[]]
 => []
 => []
 => ? ∊ {0,3,3,3,3,10,21}
[7] => [[7],[]]
 => []
 => []
 => ? ∊ {0,3,3,3,3,10,21}
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules. This statistic was discussed in [Theorem 5.7, 1].
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St000219The number of occurrences of the pattern 231 in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000379The number of Hamiltonian cycles in a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000929The constant term of the character polynomial of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.