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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000823
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St000823: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 2
{{1},{2}}
=> 1
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 1
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> 3
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 1
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 3
{{1,2,4,5},{3}}
=> 4
{{1,2,4},{3,5}}
=> 3
{{1,2,4},{3},{5}}
=> 2
{{1,2,5},{3,4}}
=> 3
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 3
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 2
{{1,3,4,5},{2}}
=> 4
{{1,3,4},{2,5}}
=> 3
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 3
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 2
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 1
Description
The number of unsplittable factors of the set partition.
Let $\pi$ be a set partition of $m$ into $k$ parts and $\sigma$ a set partition of $n$ into $\ell$ parts. Then their split product is the set partition of $m+n$ with blocks
$$ B_1 \cup (C_1+m),\dots,B_k \cup (C_k+m), C_{k+1}+m,\dots,C_\ell+m $$
if $k\leq \ell$ and
$$ B_1 \cup (C_1+m),\dots,B_\ell \cup (C_\ell+m), B_{\ell+1},\dots,B_k $$
otherwise.
Matching statistic: St000260
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(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 23%●distinct values known / distinct values provided: 12%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 23%●distinct values known / distinct values provided: 12%
Values
{{1,2}}
=> [2] => [2] => ([],2)
=> ? = 2
{{1},{2}}
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
{{1,2,3}}
=> [3] => [3] => ([],3)
=> ? = 3
{{1,2},{3}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? = 2
{{1,3},{2}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? = 2
{{1},{2,3}}
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2,3,4}}
=> [4] => [4] => ([],4)
=> ? = 4
{{1,2,3},{4}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? = 3
{{1,2,4},{3}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? = 3
{{1,2},{3,4}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
{{1,3,4},{2}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? = 3
{{1,3},{2,4}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
{{1,4},{2,3}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
{{1},{2,3,4}}
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
{{1},{2,4},{3}}
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{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
{{1,2,3,4,5}}
=> [5] => [5] => ([],5)
=> ? = 5
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? = 4
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? = 4
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? = 4
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? = 4
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,5},{2,3,4}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
{{1},{2,3,4,5}}
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2,3,4},{5}}
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1},{2,3,5},{4}}
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{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
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
{{1,4},{2,5},{3}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1},{2,4,5},{3}}
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4},{3},{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
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1},{2,5},{3,4}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,4,5}}
=> [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{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},{2,5},{3},{4}}
=> [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
{{1},{2},{3,5},{4}}
=> [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},{2},{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
{{1},{2},{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},{2,3,4,5,6}}
=> [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
{{1},{2,3,4,5},{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
{{1},{2,3,4,6},{5}}
=> [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
{{1},{2,3,4},{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
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [3,1,1,1] => ([(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)
=> 1
{{1},{2,3,5,6},{4}}
=> [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
{{1},{2,3,5},{4,6}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,3,5},{4},{6}}
=> [1,3,1,1] => [3,1,1,1] => ([(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)
=> 1
{{1},{2,3,6},{4,5}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,3},{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},{2,3},{4,5},{6}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,3,6},{4},{5}}
=> [1,3,1,1] => [3,1,1,1] => ([(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)
=> 1
{{1},{2,3},{4,6},{5}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [2,1,1,1,1] => ([(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
{{1},{2,4,5,6},{3}}
=> [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
{{1},{2,4,5},{3,6}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,4,5},{3},{6}}
=> [1,3,1,1] => [3,1,1,1] => ([(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)
=> 1
{{1},{2,4,6},{3,5}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,4},{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
{{1},{2,4},{3,5},{6}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,4,6},{3},{5}}
=> [1,3,1,1] => [3,1,1,1] => ([(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)
=> 1
{{1},{2,4},{3,6},{5}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,4},{3},{5,6}}
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,4},{3},{5},{6}}
=> [1,2,1,1,1] => [2,1,1,1,1] => ([(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
{{1},{2,5,6},{3,4}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1},{2,5},{3,4,6}}
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
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