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Your data matches 243 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [2]
=> 1
[2,1] => [1,2] => [1,2] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [4,1]
=> 2
Description
The length of the partition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [1,1]
=> 1
[2,1] => [1,2] => [1,2] => [1,1]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [2,2,1]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [2,2,1]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [2,1,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000346
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [2]
=> 1
[2,1] => [1,2] => [1,2] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [4,1]
=> 2
Description
The number of coarsenings of a partition.
A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000396
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [.,.]
=> 1
[1,2] => [1,2] => [1,2] => [.,[.,.]]
=> 1
[2,1] => [1,2] => [1,2] => [.,[.,.]]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 2
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000533
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [2]
=> 1
[2,1] => [1,2] => [1,2] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [4,1]
=> 2
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000755
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [1,1]
=> 1
[2,1] => [1,2] => [1,2] => [1,1]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [2,1,1]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [2,2,1]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [2,2,1]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [2,1,1,1]
=> 2
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St000758
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2] => [2] => 1
[2,1] => [1,2] => [2] => [2] => 1
[1,2,3] => [1,2,3] => [3] => [3] => 1
[1,3,2] => [1,2,3] => [3] => [3] => 1
[2,1,3] => [1,2,3] => [3] => [3] => 1
[2,3,1] => [1,2,3] => [3] => [3] => 1
[3,1,2] => [1,3,2] => [2,1] => [1,2] => 2
[3,2,1] => [1,3,2] => [2,1] => [1,2] => 2
[1,2,3,4] => [1,2,3,4] => [4] => [4] => 1
[1,2,4,3] => [1,2,3,4] => [4] => [4] => 1
[1,3,2,4] => [1,2,3,4] => [4] => [4] => 1
[1,3,4,2] => [1,2,3,4] => [4] => [4] => 1
[1,4,2,3] => [1,2,4,3] => [3,1] => [1,3] => 2
[1,4,3,2] => [1,2,4,3] => [3,1] => [1,3] => 2
[2,1,3,4] => [1,2,3,4] => [4] => [4] => 1
[2,1,4,3] => [1,2,3,4] => [4] => [4] => 1
[2,3,1,4] => [1,2,3,4] => [4] => [4] => 1
[2,3,4,1] => [1,2,3,4] => [4] => [4] => 1
[2,4,1,3] => [1,2,4,3] => [3,1] => [1,3] => 2
[2,4,3,1] => [1,2,4,3] => [3,1] => [1,3] => 2
[3,1,2,4] => [1,3,2,4] => [2,2] => [2,2] => 2
[3,1,4,2] => [1,3,4,2] => [3,1] => [1,3] => 2
[3,2,1,4] => [1,3,2,4] => [2,2] => [2,2] => 2
[3,2,4,1] => [1,3,4,2] => [3,1] => [1,3] => 2
[3,4,1,2] => [1,3,2,4] => [2,2] => [2,2] => 2
[3,4,2,1] => [1,3,2,4] => [2,2] => [2,2] => 2
[4,1,3,2] => [1,4,2,3] => [2,2] => [2,2] => 2
[4,2,3,1] => [1,4,2,3] => [2,2] => [2,2] => 2
[4,3,1,2] => [1,4,2,3] => [2,2] => [2,2] => 2
[4,3,2,1] => [1,4,2,3] => [2,2] => [2,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => [1,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => [1,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => [1,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => [1,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [2,3] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [1,4] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [2,3] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [1,4] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [2,3] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => [2,3] => 2
[1,5,2,4,3] => [1,2,5,3,4] => [3,2] => [2,3] => 2
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000760
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [1,1] => 1
[2,1] => [1,2] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[3,1,2] => [1,3,2] => [2,3,1] => [2,1] => 2
[3,2,1] => [1,3,2] => [2,3,1] => [2,1] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,5,2,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition.
By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
Matching statistic: St000783
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [2]
=> 1
[2,1] => [1,2] => [1,2] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,1,2] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[3,2,1] => [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [3,1]
=> 2
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,2]
=> 2
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [3,1]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [4,1]
=> 2
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St000903
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [1,1] => 1
[2,1] => [1,2] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1
[3,1,2] => [1,3,2] => [2,3,1] => [2,1] => 2
[3,2,1] => [1,3,2] => [2,3,1] => [2,1] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2
[1,5,2,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2
Description
The number of different parts of an integer composition.
The following 233 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St001432The order dimension of the partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St001092The number of distinct even parts of a partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000392The length of the longest run of ones in a binary word. St000730The maximal arc length of a set partition. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000390The number of runs of ones in a binary word. St001732The number of peaks visible from the left. St001884The number of borders of a binary word. St000183The side length of the Durfee square of an integer partition. St000225Difference between largest and smallest parts in a partition. St000292The number of ascents of a binary word. St000759The smallest missing part in an integer partition. St000897The number of different multiplicities of parts of an integer partition. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000630The length of the shortest palindromic decomposition of a binary word. St000983The length of the longest alternating subword. St001128The exponens consonantiae of a partition. St000659The number of rises of length at least 2 of a Dyck path. St000691The number of changes of a binary word. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000764The number of strong records in an integer composition. St000628The balance of a binary word. St000920The logarithmic height of a Dyck path. St000381The largest part of an integer composition. St001335The cardinality of a minimal cycle-isolating set of a graph. St000028The number of stack-sorts needed to sort a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St000701The protection number of a binary tree. St000097The order of the largest clique of the graph. St000451The length of the longest pattern of the form k 1 2. St000862The number of parts of the shifted shape of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000053The number of valleys of the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000035The number of left outer peaks of a permutation. St000679The pruning number of an ordered tree. St000996The number of exclusive left-to-right maxima of a permutation. St000647The number of big descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000098The chromatic number of a graph. St000834The number of right outer peaks of a permutation. St000352The Elizalde-Pak rank of a permutation. St000245The number of ascents of a permutation. St000058The order of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000397The Strahler number of a rooted tree. St000527The width of the poset. St001029The size of the core of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000091The descent variation of a composition. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001673The degree of asymmetry of an integer composition. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001839The number of excedances of a set partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001062The maximal size of a block of a set partition. St000640The rank of the largest boolean interval in a poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001469The holeyness of a permutation. St000485The length of the longest cycle of a permutation. St000068The number of minimal elements in a poset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001665The number of pure excedances of a permutation. St000353The number of inner valleys of a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000264The girth of a graph, which is not a tree. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000454The largest eigenvalue of a graph if it is integral. St001427The number of descents of a signed permutation. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000007The number of saliances of the permutation. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St000542The number of left-to-right-minima of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001734The lettericity of a graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000918The 2-limited packing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000387The matching number of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001271The competition number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000061The number of nodes on the left branch of a binary tree. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000456The monochromatic index of a connected graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001487The number of inner corners of a skew partition. St001555The order of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001960The number of descents of a permutation minus one if its first entry is not one. St001557The number of inversions of the second entry of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000307The number of rowmotion orbits of a poset. St001624The breadth of a lattice. St001895The oddness of a signed permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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