CyclicallyAdjacentTranspositions

Two-cycles (i, i+1), and the two-cycle (1, n).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001466via Permutation✓↗

CyclicallyAdjacentTranspositions(p)Two-cycles (i, i+1), and the two-cycle (1, n).

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.
• For n = 2 the one two-cycle is counted once.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 2 else Count(Filter(
                                       Range(1, Length(_x) - 1, 1),
                                       Function(
                                                 _x[i] == i + 1 && _x[i + 1] ==
                                                                   i,
                                                 i,
                                               ),
                                     )) + (1 if Length(_x) > 2 && _x[1] ==
                                                                 Length(_x) && _x[Length(_x)] ==
                                                                     1 else 0)

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{CyclicallyAdjacentTranspositions}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{CyclicallyAdjacentTranspositions}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001466

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{CyclicallyAdjacentTranspositions})$Out$0$

Known value · FindStat St001466

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001466

About

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