DistinctDiagonalSums

The number of distinct sums along the diagonals of the permutation matrix.

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DistinctDiagonalSums(p)The number of distinct sums along the diagonals of the permutation matrix.

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.
• The sum along the diagonal k counts the i with p(i) - i = k, empty diagonals included.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000891

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000891

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000891

About

Examples	5
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