Identifier
Values
[(1,3),(2,4)] => [3,4,1,2] => [2,1,1] => 011 => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [2,2,1,1] => 0011 => 1
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [4,1,1] => 011 => 1
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,1,1] => 011 => 1
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [4,1,1] => 011 => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [2,2,1,1] => 0011 => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,2,1,1] => 0011 => 1
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [4,1,1] => 011 => 1
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,1,1] => 011 => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [4,1,1] => 011 => 1
[(1,5),(2,3),(4,6),(7,8)] => [5,3,2,6,1,4,8,7] => [4,2,1,1] => 0011 => 1
[(1,7),(2,3),(4,5),(6,8)] => [7,3,2,5,4,8,1,6] => [4,2,1,1] => 0011 => 1
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [4,2,1,1] => 0011 => 1
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [4,2,1,1] => 0011 => 1
[(1,5),(2,4),(3,6),(7,8)] => [5,4,6,2,1,3,8,7] => [4,2,1,1] => 0011 => 1
[(1,3),(2,6),(4,5),(7,8)] => [3,6,1,5,4,2,8,7] => [4,2,1,1] => 0011 => 1
[(1,4),(2,6),(3,5),(7,8)] => [4,6,5,1,3,2,8,7] => [4,2,1,1] => 0011 => 1
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [4,2,1,1] => 0011 => 1
[(1,7),(2,5),(3,4),(6,8)] => [7,5,4,3,2,8,1,6] => [6,1,1] => 011 => 1
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [6,1,1] => 011 => 1
[(1,7),(2,6),(3,4),(5,8)] => [7,6,4,3,8,2,1,5] => [6,1,1] => 011 => 1
[(1,5),(2,7),(3,4),(6,8)] => [5,7,4,3,1,8,2,6] => [4,2,1,1] => 0011 => 1
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [4,2,1,1] => 0011 => 1
[(1,3),(2,8),(4,5),(6,7)] => [3,8,1,5,4,7,6,2] => [4,2,1,1] => 0011 => 1
[(1,4),(2,8),(3,5),(6,7)] => [4,8,5,1,3,7,6,2] => [4,2,1,1] => 0011 => 1
[(1,5),(2,8),(3,4),(6,7)] => [5,8,4,3,1,7,6,2] => [4,3,1] => 011 => 1
[(1,6),(2,8),(3,4),(5,7)] => [6,8,4,3,7,1,5,2] => [4,2,1,1] => 0011 => 1
[(1,7),(2,8),(3,4),(5,6)] => [7,8,4,3,6,5,1,2] => [4,2,1,1] => 0011 => 1
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [6,1,1] => 011 => 1
[(1,6),(2,8),(3,5),(4,7)] => [6,8,5,7,3,1,4,2] => [4,2,1,1] => 0011 => 1
[(1,4),(2,8),(3,6),(5,7)] => [4,8,6,1,7,3,5,2] => [4,2,1,1] => 0011 => 1
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [4,2,1,1] => 0011 => 1
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [4,2,1,1] => 0011 => 1
[(1,4),(2,7),(3,6),(5,8)] => [4,7,6,1,8,3,2,5] => [4,2,1,1] => 0011 => 1
[(1,7),(2,6),(3,5),(4,8)] => [7,6,5,8,3,2,1,4] => [6,1,1] => 011 => 1
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [6,1,1] => 011 => 1
[(1,5),(2,4),(3,7),(6,8)] => [5,4,7,2,1,8,3,6] => [4,2,1,1] => 0011 => 1
[(1,7),(2,4),(3,6),(5,8)] => [7,4,6,2,8,3,1,5] => [4,2,1,1] => 0011 => 1
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [4,2,1,1] => 0011 => 1
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [4,2,1,1] => 0011 => 1
[(1,7),(2,3),(4,6),(5,8)] => [7,3,2,6,8,4,1,5] => [4,2,1,1] => 0011 => 1
[(1,5),(2,3),(4,7),(6,8)] => [5,3,2,7,1,8,4,6] => [4,2,1,1] => 0011 => 1
[(1,4),(2,3),(5,7),(6,8)] => [4,3,2,1,7,8,5,6] => [4,2,1,1] => 0011 => 1
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,8,7,6,5] => [4,2,1,1] => 0011 => 1
[(1,5),(2,3),(4,8),(6,7)] => [5,3,2,8,1,7,6,4] => [4,3,1] => 011 => 1
[(1,6),(2,3),(4,8),(5,7)] => [6,3,2,8,7,1,5,4] => [4,3,1] => 011 => 1
[(1,7),(2,3),(4,8),(5,6)] => [7,3,2,8,6,5,1,4] => [4,3,1] => 011 => 1
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [6,1,1] => 011 => 1
[(1,7),(2,4),(3,8),(5,6)] => [7,4,8,2,6,5,1,3] => [4,2,1,1] => 0011 => 1
[(1,6),(2,4),(3,8),(5,7)] => [6,4,8,2,7,1,5,3] => [4,2,1,1] => 0011 => 1
[(1,5),(2,4),(3,8),(6,7)] => [5,4,8,2,1,7,6,3] => [4,3,1] => 011 => 1
[(1,3),(2,5),(4,8),(6,7)] => [3,5,1,8,2,7,6,4] => [4,2,1,1] => 0011 => 1
[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,8,3,7,6,4] => [4,2,1,1] => 0011 => 1
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,8,7,3,5,4] => [4,2,1,1] => 0011 => 1
[(1,3),(2,6),(4,8),(5,7)] => [3,6,1,8,7,2,5,4] => [4,2,1,1] => 0011 => 1
[(1,7),(2,5),(3,8),(4,6)] => [7,5,8,6,2,4,1,3] => [4,2,1,1] => 0011 => 1
[(1,8),(2,5),(3,7),(4,6)] => [8,5,7,6,2,4,3,1] => [6,1,1] => 011 => 1
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => [6,1,1] => 011 => 1
[(1,7),(2,6),(3,8),(4,5)] => [7,6,8,5,4,2,1,3] => [6,1,1] => 011 => 1
[(1,3),(2,7),(4,8),(5,6)] => [3,7,1,8,6,5,2,4] => [4,2,1,1] => 0011 => 1
[(1,2),(3,7),(4,8),(5,6)] => [2,1,7,8,6,5,3,4] => [4,2,1,1] => 0011 => 1
[(1,3),(2,8),(4,7),(5,6)] => [3,8,1,7,6,5,4,2] => [6,1,1] => 011 => 1
[(1,4),(2,8),(3,7),(5,6)] => [4,8,7,1,6,5,3,2] => [6,1,1] => 011 => 1
[(1,5),(2,8),(3,7),(4,6)] => [5,8,7,6,1,4,3,2] => [6,1,1] => 011 => 1
[(1,6),(2,8),(3,7),(4,5)] => [6,8,7,5,4,1,3,2] => [6,1,1] => 011 => 1
[(1,7),(2,8),(3,6),(4,5)] => [7,8,6,5,4,3,1,2] => [6,1,1] => 011 => 1
[(1,9),(2,7),(3,6),(4,5),(8,10)] => [9,7,6,5,4,3,2,10,1,8] => [8,1,1] => 011 => 1
[(1,9),(2,10),(3,8),(4,7),(5,6)] => [9,10,8,7,6,5,4,3,1,2] => [8,1,1] => 011 => 1
search for individual values
searching the database for the individual values of this statistic
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
odd parts
Description
Return the binary word indicating which parts of the partition are odd.