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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000586
St000586: 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}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 2
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 2
{{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}}
=> 3
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 4
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 3
{{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}}
=> 1
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
Matching statistic: St000833
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [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] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,4,5,3] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,2,3,5] => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,5,2,3] => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,4,5,2,3] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,3,5,2,4] => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2
Description
The comajor index of a permutation.
This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Matching statistic: St001811
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [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] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [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] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,4,3,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001843
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001843: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001843: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => {{1},{2}}
=> 0
{{1},{2}}
=> [1,2] => [2,1] => {{1,2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => {{1},{2,3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => {{1,2},{3}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => {{1,3},{2}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 3
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1
Description
The Z-index of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The Z-index of $w$ equals
$$
\sum_{i < j} w_{i,j},
$$
where $w_{i,j}$ is the word obtained from $w$ by removing all letters different from $i$ and $j$.
Matching statistic: St001313
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => 0 => 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => 0 => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 00 => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 00 => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 01 => 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 00 => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 00 => 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 001 => 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 001 => 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 010 => 2 = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 010 => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 010 => 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 010 => 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 001 => 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 000 => 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 4 = 3 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 0001 => 4 = 3 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 0010 => 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 3 = 2 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 0010 => 3 = 2 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 4 = 3 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 0001 => 4 = 3 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 0010 => 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 0010 => 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 0010 => 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 0100 => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 0100 => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 0010 => 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 0101 => 5 = 4 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 0100 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 0010 => 3 = 2 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 0100 => 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => 0100 => 2 = 1 + 1
Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St001596
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 60% ●values known / values provided: 85%●distinct values known / distinct values provided: 60%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 60% ●values known / values provided: 85%●distinct values known / distinct values provided: 60%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [[2],[]]
=> 0
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [[1,1],[]]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {2,3,3,3,3,3,3,3,4,4,4}
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001176
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Values
{{1,2}}
=> [2]
=> [2]
=> []
=> ? = 0
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> 0
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,2,2}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,2,2}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,2,2}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{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
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{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]
=> 2
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001714
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Values
{{1,2}}
=> [2]
=> [2]
=> []
=> ? = 0
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> 0
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,2,2}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,2,2}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,2,2}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 0
{{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
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{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]
=> 2
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4}
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000771
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 80%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 80%
Values
{{1,2}}
=> [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,2,2} + 1
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,2,2} + 1
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,2,2} + 1
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,2,2} + 1
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2} + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,2,3,4,5}}
=> [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,4,5},{2,3}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,4},{2,3,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2},{3,4}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4} + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St001964
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
{{1,2}}
=> [2,1] => [1,2] => ([(0,1)],2)
=> 0
{{1},{2}}
=> [1,2] => [2,1] => ([],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 0
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,1}
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,1}
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => ([],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,2,2}
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {0,1,1,1,2,2}
{{1,3,4},{2}}
=> [3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,2,2}
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,1,1,1,2,2}
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001867The number of alignments of type EN of a signed permutation. St001438The number of missing boxes of a skew partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001864The number of excedances of a signed permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000454The largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001875The number of simple modules with projective dimension at most 1. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001060The distinguishing index of a graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000993The multiplicity of the largest part of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001857The number of edges in the reduced word graph of a signed permutation. St001623The number of doubly irreducible elements of a lattice.
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