HalfPermanentOfIdentityPlusPermutation

Half the permanent of the identity matrix plus the permutation matrix, which is 2 to the power of the cycles less one.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001661via Permutation✓↗

HalfPermanentOfIdentityPlusPermutation(p)Half the permanent of the identity matrix plus the permutation matrix, which is 2 to the power of the cycles less one.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation 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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else 2 ^ (Count(Filter(
                                            Range(1, Length(_x), 1),
                                            Function(
                                                      Fold(
                                                            Function(
                                                                      w if w <=
                                                                           0 else 0 if w == i else -1 if w < i else _x[w],
                                                                      w,
                                                                      k,
                                                                    ),
                                                            _x[i],
                                                            Range(
                                                                   1,
                                                                   Length(_x),
                                                                   1,
                                                                 ),
                                                          ) == 0,
                                                      i,
                                                    ),
                                          )) - 1)

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{HalfPermanentOfIdentityPlusPermutation}({\operatorname{Permutation}}([3, 1, 2]))$Out$1$

the one-line word $2413$

live ↗

In$\mathrm{HalfPermanentOfIdentityPlusPermutation}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$1$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001661

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{HalfPermanentOfIdentityPlusPermutation})$Out$2$

Known value · FindStat St001661

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{HalfPermanentOfIdentityPlusPermutation})$Out$1$

Known value · FindStat St001661

About

Examples	5
Crosswalk	FindStat