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Your data matches 186 different statistics following compositions of up to 3 maps.
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Matching statistic: St001743
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> 0
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 1
([(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
Description
The discrepancy of a graph.
For a subset $C$ of the set of vertices $V(G)$, and a vertex $v$, let $d_{C, v} = |\#(N(v)\cap C) - \#(N(v)\cap(V\setminus C))|$, and let $d_C$ be the maximal value of $d_{C, v}$ over all vertices.
Then the discrepancy of the graph is the minimal value of $d_C$ over all subsets of $V(G)$.
Graphs with at most $8$ vertices have discrepancy at most $2$, but there are graphs with arbitrary discrepancy.
Matching statistic: St000260
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1] => [1] => ([],1)
=> 0
([],2)
=> [2] => [2] => ([],2)
=> ? = 0
([(0,1)],2)
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => [3] => ([],3)
=> ? ∊ {0,2}
([(1,2)],3)
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,2}
([],4)
=> [4] => [4] => ([],4)
=> ? ∊ {0,0,1,1,2}
([(2,3)],4)
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,2}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,2}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,2}
([],5)
=> [5] => [5] => ([],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2}
([],6)
=> [6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
([(3,5),(4,5)],6)
=> [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,5),(3,4)],6)
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(3,4),(3,5),(4,5)],6)
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000698
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,1}
([],3)
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {0,1,2}
([(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(1,2)],3)
=> [3]
=> [3]
=> []
=> ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [3]
=> []
=> ? ∊ {0,1,2}
([],4)
=> [1,1,1,1]
=> [2,2]
=> [2]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,2}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,2}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,2}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [2,2,1]
=> [2,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [4,1]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
([],6)
=> [1,1,1,1,1,1]
=> [2,2,2]
=> [2,2]
=> 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [3,2,1]
=> [2,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [3,2,1]
=> [2,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [3,3]
=> [3]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001603
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 71%●distinct values known / distinct values provided: 33%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 71%●distinct values known / distinct values provided: 33%
Values
([],1)
=> ([],1)
=> [1]
=> ? = 0
([],2)
=> ([],1)
=> [1]
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1}
([],3)
=> ([],1)
=> [1]
=> ? ∊ {0,1,2}
([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1,2}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 71%●distinct values known / distinct values provided: 33%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 71%●distinct values known / distinct values provided: 33%
Values
([],1)
=> ([],1)
=> [1]
=> ? = 0
([],2)
=> ([],1)
=> [1]
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1}
([],3)
=> ([],1)
=> [1]
=> ? ∊ {0,1,2}
([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1,2}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 71%●distinct values known / distinct values provided: 33%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 71%●distinct values known / distinct values provided: 33%
Values
([],1)
=> ([],1)
=> [1]
=> ? = 0
([],2)
=> ([],1)
=> [1]
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1}
([],3)
=> ([],1)
=> [1]
=> ? ∊ {0,1,2}
([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1,2}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],5)
=> ([],1)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([],6)
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001629
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
([],1)
=> ([],1)
=> [1] => [1] => ? = 0
([],2)
=> ([],1)
=> [1] => [1] => ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1}
([],3)
=> ([],1)
=> [1] => [1] => ? ∊ {0,1,2}
([(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,2}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([],4)
=> ([],1)
=> [1] => [1] => ? ∊ {0,0,1,1,1,1,2}
([(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => 1
([],5)
=> ([],1)
=> [1] => [1] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,1] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 1
([],6)
=> ([],1)
=> [1] => [1] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1,1] => [2] => ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,1] => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,1] => 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000284
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 70%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 70%●distinct values known / distinct values provided: 33%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([],1)
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1}
([],3)
=> ([],1)
=> []
=> ? ∊ {0,1,2}
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1,2}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> []
=> ? ∊ {0,0,1,1,1,1,2}
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([],5)
=> ([],1)
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 1
([],6)
=> ([],1)
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000704
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 70%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 70%●distinct values known / distinct values provided: 33%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([],1)
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1}
([],3)
=> ([],1)
=> []
=> ? ∊ {0,1,2}
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1,2}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> []
=> ? ∊ {0,0,1,1,1,1,2}
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([],5)
=> ([],1)
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 1
([],6)
=> ([],1)
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000706
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 70%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 70%●distinct values known / distinct values provided: 33%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([],1)
=> []
=> ? ∊ {0,1}
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1}
([],3)
=> ([],1)
=> []
=> ? ∊ {0,1,2}
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1,2}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,1,2}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> []
=> ? ∊ {0,0,1,1,1,1,2}
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,1,2}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([],5)
=> ([],1)
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 1
([],6)
=> ([],1)
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
Description
The product of the factorials of the multiplicities of an integer partition.
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St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000659The number of rises of length at least 2 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000535The rank-width of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001340The cardinality of a minimal non-edge isolating set of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001393The induced matching number of a graph. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001273The projective dimension of the first term in an injective coresolution of the regular module. 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. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000455The second largest eigenvalue of a graph if it is integral. St001592The maximal number of simple paths between any two different vertices of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000387The matching number of a graph. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001271The competition number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000741The Colin de Verdière graph invariant. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000120The number of left tunnels of a Dyck path. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000088The row sums of the character table of the symmetric group. St000159The number of distinct parts of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001933The largest multiplicity of a part in an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001060The distinguishing index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition.
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