Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St000907
(load all 2 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => 3
[[1],[2]]
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[1],[2],[3]]
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => 2
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000469
(load all 2 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000469: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
Description
The distinguishing number of a graph. This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring. For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].
Matching statistic: St000723
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
Description
The maximal cardinality of a set of vertices with the same neighbourhood in a graph. The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].
Matching statistic: St000776
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000776: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
Description
The maximal multiplicity of an eigenvalue in a graph.
Matching statistic: St000835
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000835: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => [6,1]
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
Description
The minimal difference in size when partitioning the integer partition into two subpartitions. This is the optimal value of the optimisation version of the partition problem [1].
Matching statistic: St000986
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St000992
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => [6,1]
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
Description
The alternating sum of the parts of an integer partition. For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St001055
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001055: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => [6,1]
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
Description
The Grundy value for the game of removing cells of a row in an integer partition. Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses.
Matching statistic: St001366
(load all 2 compositions to match this statistic)
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001366: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
Description
The maximal multiplicity of a degree of a vertex of a graph.
Matching statistic: St001691
Mapping
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001691: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[3]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 4
[[1,4],[4]]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(5,6)],7)
 => 5
[[1],[2],[4]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1],[3],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[3],[4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 3
[[1,1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 4
[[1,1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
Description
The number of kings in a graph. A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000982The length of the longest constant subword. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001933The largest multiplicity of a part in an integer partition. St000626The minimal period of a binary word. St001091The number of parts in an integer partition whose next smaller part has the same size. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000315The number of isolated vertices of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000674The number of hills of a Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.