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Matching statistic: St001693
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Mapping
St001693: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => 1
{{1,2}}
 => 1
{{1},{2}}
 => 0
{{1,2,3}}
 => 1
{{1,2},{3}}
 => 1
{{1,3},{2}}
 => 0
{{1},{2,3}}
 => 1
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 1
{{1,2,3},{4}}
 => 1
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 2
{{1,2},{3},{4}}
 => 1
{{1,3,4},{2}}
 => 1
{{1,3},{2,4}}
 => 2
{{1,3},{2},{4}}
 => 0
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 1
{{1},{2,3},{4}}
 => 1
{{1,4},{2},{3}}
 => 0
{{1},{2,4},{3}}
 => 0
{{1},{2},{3,4}}
 => 1
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 1
{{1,2,3,4},{5}}
 => 1
{{1,2,3,5},{4}}
 => 1
{{1,2,3},{4,5}}
 => 2
{{1,2,3},{4},{5}}
 => 1
{{1,2,4,5},{3}}
 => 1
{{1,2,4},{3,5}}
 => 2
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 2
{{1,2},{3,4,5}}
 => 2
{{1,2},{3,4},{5}}
 => 2
{{1,2,5},{3},{4}}
 => 1
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 2
{{1,2},{3},{4},{5}}
 => 1
{{1,3,4,5},{2}}
 => 1
{{1,3,4},{2,5}}
 => 2
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => 2
{{1,3},{2,4},{5}}
 => 2
{{1,3,5},{2},{4}}
 => 0
{{1,3},{2,5},{4}}
 => 1
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 0
{{1,4,5},{2,3}}
 => 2
{{1,4},{2,3,5}}
 => 2
Description
The excess length of a longest path consisting of elements and blocks of a set partition. Let $p$ be a set partition of $\{1,\dots,n\}$. Let $G$ be the graph with edges $(i,i+1)$ for $i\in\{1,\dots,n-1\}$ and $(i, b)$, whenever $i$ is an element of a non-singleton block $b\in p$. Then this statistic records the length of the longest path from $1$ to $n$ in $G$, reduced by $n$. Conjecturally, a longest path has more than $n$ vertices provided that the set partition has no singletons.