Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000274
Mapping
St000274: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 0
([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => 0
([(2,3)],4)
 => 0
([(1,3),(2,3)],4)
 => 0
([(0,3),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2)],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => 0
([(3,4)],5)
 => 0
([(2,4),(3,4)],5)
 => 0
([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(1,4),(2,3)],5)
 => 0
([(1,4),(2,3),(3,4)],5)
 => 0
([(0,1),(2,4),(3,4)],5)
 => 0
([(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
Description
The number of perfect matchings of a graph. A matching of a graph $G$ is a subset $F \subset E(G)$ such that no two edges in $F$ share a vertex in common. A perfect matching $F'$ is then a matching such that every vertex in $V(G)$ is incident with exactly one edge in $F'$.
Matching statistic: St000319
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001280
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001392
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001541
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Matching statistic: St001587
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Matching statistic: St001657
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
The number of twos in an integer partition. The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St001918
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 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)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.
Matching statistic: St000207
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [1]
 => []
 => ?
 => ? = 0 + 1
([],2)
 => [1,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => ?
 => ? = 1 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ?
 => ? = 0 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ?
 => ? = 3 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ?
 => ? = 0 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0 + 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [6]
 => []
 => ?
 => ? = 0 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => []
 => ? = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => 1 = 0 + 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition.