Your data matches 377 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001175
(load all 9 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => [2]
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [2]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [2]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => 0 = 1 - 1
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: St000321
(load all 2 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => []
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => 1
([(1,3),(2,3)],4)
 => [2]
 => []
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => 1
([(2,4),(3,4)],5)
 => [2]
 => []
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => 1
([(3,5),(4,5)],6)
 => [2]
 => []
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1
Description
The number of integer partitions of n that are dominated by an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
Matching statistic: St000345
(load all 2 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => []
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => 1
([(1,3),(2,3)],4)
 => [2]
 => []
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => 1
([(2,4),(3,4)],5)
 => [2]
 => []
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => 1
([(3,5),(4,5)],6)
 => [2]
 => []
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1
Description
The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
(load all 2 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => []
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => 1
([(1,3),(2,3)],4)
 => [2]
 => []
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => 1
([(2,4),(3,4)],5)
 => [2]
 => []
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => 1
([(3,5),(4,5)],6)
 => [2]
 => []
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1
Description
The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St000346
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => []
 => []
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => 1
([(1,3),(2,3)],4)
 => [2]
 => []
 => []
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => []
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => 1
([(2,4),(3,4)],5)
 => [2]
 => []
 => []
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => []
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => []
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => []
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => []
 => 1
([(3,5),(4,5)],6)
 => [2]
 => []
 => []
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => []
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => []
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1
Description
The number of coarsenings of a partition. A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St001387
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001387: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => []
 => []
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => 1
([(1,3),(2,3)],4)
 => [2]
 => []
 => []
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => []
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => 1
([(2,4),(3,4)],5)
 => [2]
 => []
 => []
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => []
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => []
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => []
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => []
 => 1
([(3,5),(4,5)],6)
 => [2]
 => []
 => []
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => []
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => []
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1
Description
Number of standard Young tableaux of the skew shape tracing the border of the given partition. Let $\lambda \vdash n$ be a diagram with the given partition as shape. Add $n$ additional boxes, one in each column $1,\dotsc,n$, and let this be $\mu$. The statistic is the number of standard Young tableaux of skew shape $\mu/\lambda$, which is equal to $\frac{n!}{\prod_{i} (\mu_i - \lambda_i)!}$. For example, $\lambda=[2,1,1]$ gives $\mu = [4,2,1,1]$. The first row in the skew shape $\mu/\lambda$ has two boxes, so the number of SYT of shape $\mu/\lambda$ is then $4!/2 = 12$. This statistic shows up in the study of skew specialized Macdonald polynomials, where a type of charge statistic give rise to a $q$-analogue of the above formula.
Matching statistic: St001389
(load all 2 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000377
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => []
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001091
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => []
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => [1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 0 = 1 - 1
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001251
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ 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
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => [1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [2,1]
 => [1]
 => 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
Description
The number of parts of a partition that are not congruent 1 modulo 3.
The following 367 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000142The number of even parts of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000260The radius of a connected graph. St000379The number of Hamiltonian cycles in a graph. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000052The number of valleys of a Dyck path not on the x-axis. St000013The height of a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St001586The number of odd parts smaller than the largest even part in an integer partition. St000295The length of the border of a binary word. St000691The number of changes of a binary word. St000456The monochromatic index of a connected graph. 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. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000455The second largest eigenvalue of a graph if it is integral. St001371The length of the longest Yamanouchi prefix of a binary word. St001141The number of occurrences of hills of size 3 in a Dyck path. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001330The hat guessing number of a graph. St000386The number of factors DDU in a Dyck path. St000017The number of inversions of a standard tableau. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. 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. St000264The girth of a graph, which is not a tree. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. 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. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001596The number of two-by-two squares inside a skew partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001309The number of four-cliques in a graph. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001307The number of induced stars on four vertices 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. 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. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000273The domination number of a graph. St000544The cop number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St000258The burning number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000918The 2-limited packing number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000443The number of long tunnels of a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number 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. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000388The number of orbits of vertices of a graph under automorphisms. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001352The number of internal nodes in the modular decomposition of 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. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000552The number of cut vertices of a graph. St001305The number of induced cycles on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a 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. 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. 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. 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. 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 $eA$. St001316The domatic number of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001367The smallest number which does not occur as degree of a vertex in a graph. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001570The minimal number of edges to add to make a graph Hamiltonian. St001964The interval resolution global dimension of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000069The number of maximal elements of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001335The cardinality of a minimal cycle-isolating set 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. St001618The cardinality of the Frattini sublattice of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St001651The Frankl number of a lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000618The number of self-evacuating tableaux of given shape. 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. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000286The number of connected components of the complement of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001871The number of triconnected components of a graph. St000259The diameter of a connected graph. St000322The skewness of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001119The length of a shortest maximal path in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000056The decomposition (or block) number of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000349The number of different adjacency matrices of a graph. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000917The open packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001057The Grundy value of the game of creating an independent set in a graph. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001271The competition number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number 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. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001393The induced matching number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001512The minimum rank of a graph. St001590The crossing number of a perfect matching. St001642The Prague dimension of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001672The restrained domination number of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001917The order of toric promotion on the set of labellings of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000299The number of nonisomorphic vertex-induced subtrees. St000312The number of leaves in a graph. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000403The Szeged index minus the Wiener index of a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000542The number of left-to-right-minima of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000750The number of occurrences of the pattern 4213 in a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001093The detour number of a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001261The Castelnuovo-Mumford regularity of a graph. St001306The number of induced paths on four vertices in a graph. St001308The number of induced paths on three vertices in 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. 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. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001374The Padmakar-Ivan index of a graph. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001479The number of bridges of a graph. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001648The number of edges that can be added without increasing the chromatic number. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001691The number of kings in a graph. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001764The number of non-convex subsets of vertices in 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. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000268The number of strongly connected orientations of a graph. St000344The number of strongly connected outdegree sequences of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001341The number of edges in the center of a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001797The number of overfull subgraphs of a graph. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001626The number of maximal proper sublattices of a lattice. 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001060The distinguishing index of a graph. St001720The minimal length of a chain of small intervals in a lattice. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000478Another weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees.