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Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000563
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000563: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 0
{{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 0
{{1,3,5},{2,4}}
=> 0
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 0
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 0
Description
The number of overlapping pairs of blocks of a set partition.
This is also the number of occurrences of the pattern {{1,3},{2,4}} such that 1,2 are minimal and 3,4 are maximal.
Matching statistic: St000232
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 0
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000233
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 1
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000559
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000559: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000559: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.
Matching statistic: St001604
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,6},{3,4,5}}
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [3,3]
=> [3]
=> 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000938
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 75%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 75%
Values
{{1,2}}
=> [2]
=> [2]
=> []
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> [2,2,2]
=> [2,2]
=> 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [4,2]
=> [2]
=> 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [4,2]
=> [2]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [4,2]
=> [2]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3}
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000566
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000621
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 63%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000941
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Values
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0}
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0}
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1}
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 0
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3}
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001490The number of connected components of a skew partition. St001513The number of nested exceedences of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001549The number of restricted non-inversions between exceedances. St000741The Colin de Verdière graph invariant. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000454The largest eigenvalue of a graph if it is integral. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001845The number of join irreducibles minus the rank of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000068The number of minimal elements in a poset. St001866The nesting alignments of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001867The number of alignments of type EN of a signed permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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