- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000160: Integer partitions ⟶ ℤ

Values

[1,0] => [] => 0

[1,0,1,0] => [1] => 1

[1,1,0,0] => [] => 0

[1,0,1,0,1,0] => [2,1] => 1

[1,0,1,1,0,0] => [1,1] => 2

[1,1,0,0,1,0] => [2] => 1

[1,1,0,1,0,0] => [1] => 1

[1,1,1,0,0,0] => [] => 0

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

[1,0,1,0,1,1,0,0] => [2,2,1] => 1

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

[1,0,1,1,0,1,0,0] => [2,1,1] => 2

[1,0,1,1,1,0,0,0] => [1,1,1] => 3

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

[1,1,0,0,1,1,0,0] => [2,2] => 2

[1,1,0,1,0,0,1,0] => [3,1] => 1

[1,1,0,1,0,1,0,0] => [2,1] => 1

[1,1,0,1,1,0,0,0] => [1,1] => 2

[1,1,1,0,0,0,1,0] => [3] => 1

[1,1,1,0,0,1,0,0] => [2] => 1

[1,1,1,0,1,0,0,0] => [1] => 1

[1,1,1,1,0,0,0,0] => [] => 0

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

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

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 1

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

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 1

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

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

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 2

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

[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 2

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 3

[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 3

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

[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 4

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

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

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 2

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

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

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 1

[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 1

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 1

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

[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 1

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 2

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

[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 2

[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 3

[1,1,1,0,0,0,1,0,1,0] => [4,3] => 1

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

[1,1,1,0,0,1,0,0,1,0] => [4,2] => 1

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

[1,1,1,0,0,1,1,0,0,0] => [2,2] => 2

[1,1,1,0,1,0,0,0,1,0] => [4,1] => 1

[1,1,1,0,1,0,0,1,0,0] => [3,1] => 1

[1,1,1,0,1,0,1,0,0,0] => [2,1] => 1

[1,1,1,0,1,1,0,0,0,0] => [1,1] => 2

[1,1,1,1,0,0,0,0,1,0] => [4] => 1

[1,1,1,1,0,0,0,1,0,0] => [3] => 1

[1,1,1,1,0,0,1,0,0,0] => [2] => 1

[1,1,1,1,0,1,0,0,0,0] => [1] => 1

[1,1,1,1,1,0,0,0,0,0] => [] => 0

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

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

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 1

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

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 1

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 1

[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => 1

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

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

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 1

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

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 1

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

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

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 2

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

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => 2

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

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

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 2

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

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => 3

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

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

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

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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

Description

The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.