- FindStat

Identifier

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤ (values match St000320The dinv adjustment of an integer partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2096 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

LLPS

Description

The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.