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Your data matches 435 different statistics following compositions of up to 3 maps.
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Matching statistic: St001175
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(load all 3 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 0
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000017
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
Description
The number of inversions of a standard tableau.
Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
Matching statistic: St000142
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
Description
The number of even parts of a partition.
Matching statistic: St000473
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000931
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001033
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001251
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 0
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001596
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [[3,1],[]]
=> 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
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.
The following 425 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001657The number of twos in an integer partition. St000024The number of double up and double down steps of a Dyck path. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000442The maximal area to the right of an up step of a Dyck path. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000013The height of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001256Number of simple reflexive modules that are 2-stable reflexive. St000649The number of 3-excedences of a permutation. St001513The number of nested exceedences of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001644The dimension of a graph. St000098The chromatic number of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000481The number of upper covers of a partition in dominance order. St000048The multinomial of the parts of a partition. St000346The number of coarsenings of a partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000480The number of lower covers of a partition in dominance order. St001092The number of distinct even parts of a partition. St000403The Szeged index minus the Wiener index of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000544The cop number of a graph. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000311The number of vertices of odd degree in a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000350The sum of the vertex degrees of a graph. St000351The determinant of the adjacency matrix of a graph. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000387The matching number of a graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000465The first Zagreb index of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000571The F-index (or forgotten topological index) of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001309The number of four-cliques in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center 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. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001649The length of a longest trail in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001743The discrepancy of a graph. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001871The number of triconnected components of a graph. St001962The proper pathwidth of a graph. St000086The number of subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000172The Grundy number of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000268The number of strongly connected orientations of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000363The number of minimal vertex covers of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000722The number of different neighbourhoods in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000972The composition number of a graph. St001029The size of the core of a graph. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001303The number of dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001386The number of prime labellings of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001546The number of monomials in the Tutte polynomial of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001580The acyclic chromatic number of a graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001963The tree-depth of a graph. St001172The number of 1-rises at odd height of a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000455The second largest eigenvalue of a graph if it is integral. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000386The number of factors DDU in a Dyck path. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000759The smallest missing part in an integer partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000475The number of parts equal to 1 in a partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St001593This is the number of standard Young tableaux of the given shifted shape. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001556The number of inversions of the third entry of a permutation. St000929The constant term of the character polynomial of an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St000096The number of spanning trees of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000315The number of isolated vertices of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000264The girth of a graph, which is not a tree. St001890The maximum magnitude of the Möbius function of a poset. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001060The distinguishing index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000617The number of global maxima of a Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000993The multiplicity of the largest part of an integer partition. St001696The natural major index of a standard Young tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001691The number of kings in a graph. St000765The number of weak records in an integer composition. St000258The burning number of a graph. St001672The restrained domination number of a graph. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000805The number of peaks of the associated bargraph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000312The number of leaves in a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000917The open packing number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001335The cardinality of a minimal cycle-isolating set of a graph. St001571The Cartan determinant of the integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001621The number of atoms of a lattice. 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 $eA$. St001964The interval resolution global dimension of a poset. St001118The acyclic chromatic index of a graph. St000117The number of centered tunnels of a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001204Call 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 special CNakayama algebra. St001637The number of (upper) dissectors of a poset. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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 $eA$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000842The breadth of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
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