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Your data matches 68 different statistics following compositions of up to 3 maps.
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Matching statistic: St000150
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 0
([(1,2)],3)
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
([(3,4)],5)
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(4,5)],6)
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> 0
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000257
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 0
([(1,2)],3)
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([(2,3)],4)
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
([(3,4)],5)
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(4,5)],6)
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> 0
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000142
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1]
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [3]
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [5]
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [5]
=> 0
([(4,5)],6)
=> [1]
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [3]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [5]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [3]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [3,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [5]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [5]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [4]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [5]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [5]
=> 0
Description
The number of even parts of a partition.
Matching statistic: St000149
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1]
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [1]
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [2,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2,2]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000256
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1]
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [1]
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [2,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [2,2]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St001092
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1]
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [3]
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [5]
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [5]
=> 0
([(4,5)],6)
=> [1]
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [3]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [5]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [3]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [3]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [3,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [5]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [5]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [4]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [5]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [5]
=> 0
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St000052
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000386
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001021
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001021: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001021: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
Description
Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001089
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
Description
Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000455The second largest eigenvalue of a graph if it is integral. St001845The number of join irreducibles minus the rank of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. 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. St000264The girth of a graph, which is not a tree. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000916The packing number of a graph. St000917The open packing number of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001871The number of triconnected components of a graph.
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