- FindStat

Identifier

Mp00259: Graphs —vertex addition⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤ

Values

([],1) => ([],2) => [1,1] => 2

([],2) => ([],3) => [1,1,1] => 3

([(0,1)],2) => ([(1,2)],3) => [2,1] => 1

([],3) => ([],4) => [1,1,1,1] => 4

([(1,2)],3) => ([(2,3)],4) => [2,1,1] => 2

([(0,2),(1,2)],3) => ([(1,3),(2,3)],4) => [3,1] => 1

([(0,1),(0,2),(1,2)],3) => ([(1,2),(1,3),(2,3)],4) => [3,1] => 1

([],4) => ([],5) => [1,1,1,1,1] => 5

([(2,3)],4) => ([(3,4)],5) => [2,1,1,1] => 3

([(1,3),(2,3)],4) => ([(2,4),(3,4)],5) => [3,1,1] => 2

([(0,3),(1,3),(2,3)],4) => ([(1,4),(2,4),(3,4)],5) => [4,1] => 1

([(0,3),(1,2)],4) => ([(1,4),(2,3)],5) => [2,2,1] => 1

([(0,3),(1,2),(2,3)],4) => ([(1,4),(2,3),(3,4)],5) => [4,1] => 1

([(1,2),(1,3),(2,3)],4) => ([(2,3),(2,4),(3,4)],5) => [3,1,1] => 2

([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => 1

([(0,2),(0,3),(1,2),(1,3)],4) => ([(1,3),(1,4),(2,3),(2,4)],5) => [4,1] => 1

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => 1

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => 1

([],5) => ([],6) => [1,1,1,1,1,1] => 6

([(3,4)],5) => ([(4,5)],6) => [2,1,1,1,1] => 4

([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => [3,1,1,1] => 3

([(1,4),(2,4),(3,4)],5) => ([(2,5),(3,5),(4,5)],6) => [4,1,1] => 2

([(0,4),(1,4),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => 1

([(1,4),(2,3)],5) => ([(2,5),(3,4)],6) => [2,2,1,1] => 2

([(1,4),(2,3),(3,4)],5) => ([(2,5),(3,4),(4,5)],6) => [4,1,1] => 2

([(0,1),(2,4),(3,4)],5) => ([(1,2),(3,5),(4,5)],6) => [3,2,1] => 1

([(2,3),(2,4),(3,4)],5) => ([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => 3

([(0,4),(1,4),(2,3),(3,4)],5) => ([(1,5),(2,5),(3,4),(4,5)],6) => [5,1] => 1

([(1,4),(2,3),(2,4),(3,4)],5) => ([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => 2

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(1,3),(1,4),(2,3),(2,4)],5) => ([(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => 2

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5,1] => 1

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => 2

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [5,1] => 1

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,4),(1,3),(2,3),(2,4)],5) => ([(1,5),(2,4),(3,4),(3,5)],6) => [5,1] => 1

([(0,1),(2,3),(2,4),(3,4)],5) => ([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => 1

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5,1] => 1

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [5,1] => 1

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5,1] => 1

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => 2

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [5,1] => 1

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => [5,1] => 1

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => 1

([],6) => ([],7) => [1,1,1,1,1,1,1] => 7

([(4,5)],6) => ([(5,6)],7) => [2,1,1,1,1,1] => 5

([(3,5),(4,5)],6) => ([(4,6),(5,6)],7) => [3,1,1,1,1] => 4

([(2,5),(3,5),(4,5)],6) => ([(3,6),(4,6),(5,6)],7) => [4,1,1,1] => 3

([(1,5),(2,5),(3,5),(4,5)],6) => ([(2,6),(3,6),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6,1] => 1

([(2,5),(3,4)],6) => ([(3,6),(4,5)],7) => [2,2,1,1,1] => 3

([(2,5),(3,4),(4,5)],6) => ([(3,6),(4,5),(5,6)],7) => [4,1,1,1] => 3

([(1,2),(3,5),(4,5)],6) => ([(2,3),(4,6),(5,6)],7) => [3,2,1,1] => 2

([(3,4),(3,5),(4,5)],6) => ([(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => 4

([(1,5),(2,5),(3,4),(4,5)],6) => ([(2,6),(3,6),(4,5),(5,6)],7) => [5,1,1] => 2

([(0,1),(2,5),(3,5),(4,5)],6) => ([(1,2),(3,6),(4,6),(5,6)],7) => [4,2,1] => 1

([(2,5),(3,4),(3,5),(4,5)],6) => ([(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => 3

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [6,1] => 1

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => 1

([(2,4),(2,5),(3,4),(3,5)],6) => ([(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1,1] => 3

([(0,5),(1,5),(2,4),(3,4)],6) => ([(1,6),(2,6),(3,5),(4,5)],7) => [3,3,1] => 1

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [6,1] => 1

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => 3

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [6,1] => 1

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [6,1] => 1

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [6,1] => 1

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => 1

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,1,1] => 2

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [6,1] => 1

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [6,1] => 1

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => 1

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => 1

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [6,1] => 1

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => 1

([(0,5),(1,4),(2,3)],6) => ([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => 1

([(1,5),(2,4),(3,4),(3,5)],6) => ([(2,6),(3,5),(4,5),(4,6)],7) => [5,1,1] => 2

([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,2),(3,6),(4,5),(5,6)],7) => [4,2,1] => 1

([(1,2),(3,4),(3,5),(4,5)],6) => ([(2,3),(4,5),(4,6),(5,6)],7) => [3,2,1,1] => 2

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [6,1] => 1

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => 1

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [6,1] => 1

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [6,1] => 1

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,1,1] => 2

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => [6,1] => 1

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => 2

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7) => [6,1] => 1

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Description

The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].

Map

to partition of connected components

Description

Return the partition of the sizes of the connected components of the graph.

Map

vertex addition

Description

Adds a disconnected vertex to a graph.