- FindStat

Identifier

Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000160: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [8] => 1

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

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

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

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

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

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

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

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

[[[.,.],[.,.]],[.,.]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [8] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [8] => 1

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

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

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

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

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

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

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

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

[.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [8] => 1

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 5\ q$

$F_{4} = 14\ q$

$F_{5} = 42\ q$

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

promotion cycle type

Description

The cycle type of promotion on the linear extensions of a poset.

Map

to poset

Description

Return the poset obtained by interpreting the tree as a Hasse diagram.