NorthStepOrderingCount

The total orderings of the up steps such that the up steps after the k-th down step are never among the first k in the order.

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NorthStepOrderingCount(path)The total orderings of the up steps such that the up steps after the k-th down step are never among the first k in the order.

Typed signatures

Type	Package	Applies
(dyck_path) -> number	combinatorics	always

Domain: Dyck path statistics

Details

• Defined over DyckPath as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a DyckPath 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 coefficient of z_1^a_1 ... z_n^a_n in the product of (z_1 + ... + z_k), by Stanley's product formula: each factor is the height before a down step over the factorial of the up steps since the last one.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

1 if Length(_x) < 4 else Apply(
                                (us: list<integer>) => Product(Map(
                                                                    Function((us[k] -
                                                                              (k -
                                                                               1)) /
                                                                             Apply(
                                                                                    (t: integer) => 1 if t == 0 else Product(Range(1, t)),
                                                                                    us[1] if k == 1 else us[k] -
                                                                                                         us[k - 1],
                                                                                  ), k),
                                                                    Range(
                                                                           1,
                                                                           Length(_x) /
                                                                           2 - 1,
                                                                         ),
                                                                  )),
                                Map(
                                     Function(Filter(
                                                      Range(1, Length(_x)),
                                                      Function(_x[i] == 0, i),
                                                    )[k] - k, k),
                                     Range(1, Length(_x) / 2),
                                   ),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{NorthStepOrderingCount}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

In$\mathrm{NorthStepOrderingCount}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$1$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001786

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

live ↗

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

Known value · FindStat St001786

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

live ↗

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

Known value · FindStat St001786

About

Examples	5
Crosswalk	FindStat