MatrixPermanent

The permanent: the sum over the permutations s of the product of a(i, s(i)).

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MatrixPermanent(tableau)The permanent: the sum over the permutations s of the product of a(i, s(i)).

Typed signatures

Type	Package	Applies
(alternating_sign_matrix) -> number	combinatorics	always

Domain: Alternating sign matrix statistics

Details

• Defined over AlternatingSignMatrix as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a AlternatingSignMatrix and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• FindStat's St001260. Ryser's formula: (-1)^n times the sum over the sets S of columns of (-1)^|S| times the product over the rows of the row's sum over S. The sets are the numbers 0 .. 2^n - 1 read in binary.

Definition

Epsil, as the implementation runs it.

on AlternatingSignMatrix, over _x

0 if Length(_x) < 1 else Apply((cells: list<integer>) => Apply((side: integer) => Sum(Map(Function((-1) ^
                                                                                                   (side +
                                                                                                    Sum(Map(Function((m -
                                                                                                                      m %
                                                                                                                      2 ^
                                                                                                                      (j - 1)) /
                                                                                                                     2 ^
                                                                                                                     (j - 1) %
                                                                                                                     2, j), Range(1, Length(_x))))) *
                                                                                                   Product(Map(
                                                                                                                Function(Sum(Map(Function((m -
                                                                                                                                           m %
                                                                                                                                           2 ^
                                                                                                                                           (j - 1)) /
                                                                                                                                          2 ^
                                                                                                                                          (j - 1) %
                                                                                                                                          2 *
                                                                                                                                          cells[(i - 1) *
                                                                                                                                                side +
                                                                                                                                                j], j), Range(1, Length(_x)))), i),
                                                                                                                Range(1, Length(_x)),
                                                                                                              )), m), Range(0, 2 ^ side -
                                                                                                                  1))), Length(_x)), Flatten(_x))

Examples

Basic #

the alternating sign matrix $[[0,1,0],[1,-1,1],[0,1,0]]$

live ↗

In$\mathrm{MatrixPermanent}({\operatorname{AlternatingSignMatrix}}([[0, 1, 0], [1, -1, 1], [0, 1, 0]]))$Out$0$

Properties #

FindStat's value for [[1]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{AlternatingSignMatrix}}([[1]]), \text{MatrixPermanent})$Out$1$

Known value · FindStat St001260

FindStat's value for [[0,1],[1,0]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{AlternatingSignMatrix}}([[0, 1], [1, 0]]), \text{MatrixPermanent})$Out$1$

Known value · FindStat St001260

FindStat's value for [[1,0],[0,1]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{AlternatingSignMatrix}}([[1, 0], [0, 1]]), \text{MatrixPermanent})$Out$1$

Known value · FindStat St001260

About

Examples	4
Crosswalk	FindStat