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Your data matches 147 different statistics following compositions of up to 3 maps.
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Matching statistic: St000907
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Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> 1
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000145
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Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1],[2]]
=> ([],1)
=> [2]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> [3]
=> 2
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,1,2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4
[[1,1],[2]]
=> ([],1)
=> [2]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,1,2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5
[[1,1,1],[2]]
=> ([],1)
=> [2]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,1],[2,2]]
=> ([],1)
=> [2]
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> [2]
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> [2]
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> [2]
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> [2]
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> [3]
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> [2]
=> 1
Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000469
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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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1
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: St000531
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Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000531: 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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> 1
Description
The leading coefficient of the rook polynomial of an integer partition.
Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Matching statistic: St000723
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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1
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
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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1
Description
The maximal multiplicity of an eigenvalue in a graph.
Matching statistic: St000835
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%
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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> 1
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
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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 1
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St000992
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%
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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> 1
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
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%
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
[[1,1,1,2]]
=> ([(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
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7
[[1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 2
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> 1
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.
The following 137 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001366The maximal multiplicity of a degree of a vertex of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001691The number of kings in a graph. St001720The minimal length of a chain of small intervals in a lattice. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000013The height of a Dyck path. 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. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001093The detour number of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001523The degree of symmetry of a Dyck path. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001933The largest multiplicity of a part in an integer partition. St000259The diameter of a connected graph. St000439The position of the first down step of a Dyck path. St000626The minimal period of a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001091The number of parts in an integer partition whose next smaller part has the same size. St001955The number of natural descents for set-valued two row standard Young tableaux. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000025The number of initial rises of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001330The hat guessing number of a graph. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000454The largest eigenvalue of a graph if it is integral. St000120The number of left tunnels of a Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. 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. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001812The biclique partition number of a graph. St000080The rank of the poset. St000189The number of elements in the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000327The number of cover relations in a poset. St000674The number of hills of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001621The number of atoms of a lattice. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001118The acyclic chromatic index of a graph. St001060The distinguishing index of a graph. St000456The monochromatic index of a connected graph. 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. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000284The Plancherel distribution on integer partitions. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. 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. St000567The sum of the products of all pairs of parts. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St000997The even-odd crank of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001808The box weight or horizontal decoration of a Dyck path.
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