DescentWordFiberSize

The permutations of the same size with the same descents as p.

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DescentWordFiberSize(p)The permutations of the same size with the same descents as p.

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.
• Counted by the rank of the last entry: a descent keeps the partial sums from the right, an ascent from the left.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

1 if Length(_x) < 2 else Sum(Fold(
                                   Function(Map(
                                                 Function(Sum(Map(
                                                                   Function(
                                                                             a[m],
                                                                             m,
                                                                           ),
                                                                   Filter(
                                                                           Range(1, Count(a)),
                                                                           Function(m >= j, m),
                                                                         ),
                                                                 )) if _x[b] >
                                                                      _x[b + 1] else Sum(Map(
                                                                                     Function(a[m], m),
                                                                                     Filter(
                                                                                             Range(1, Count(a)),
                                                                                             Function(m < j, m),
                                                                                           ),
                                                                                   )), j),
                                                 Range(1, Count(a) + 1),
                                               ), a, b),
                                   [1],
                                   Range(1, Length(_x) - 1),
                                 ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $312$, as a plain list

live ↗

In$\mathrm{DescentWordFiberSize}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$2$

the one-line word $2413$

live ↗

In$\mathrm{DescentWordFiberSize}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$5$

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000530

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000530

FindStat's value for [1,2,3]

live ↗

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

Known value · FindStat St000530

About

Examples	6
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