DiagonalStandardMonomialDegree

The degree of the standard monomial of the path relative to the diagonal boundary.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001508via DyckPath✓↗

DiagonalStandardMonomialDegree(path)The degree of the standard monomial of the path relative to the diagonal boundary.

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.
• FindStat's St001508, as its Sage code walks the path. St001509, the same relative to the trivial lower boundary, is not adopted: its published code does not run.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Fold(
                               Function(
                                         ([a[1] + 1, a[2], a[3]] if b <
                                                                    (2 * a[2] -
                                                                     1 if a[2] <=
                                                                         Length(_x) /
                                                                         2 else Length(_x) +
                                                                               1) else [
                                                                            a[1],
                                                                            a[2] +
                                                                            1,
                                                                            a[3],
                                                                          ]) if _x[b] ==
                                                                               1 else [
                                                                                       a[1] - 1,
                                                                                       a[2] + 1,
                                                                                       a[3] + 1,
                                                                                     ] if a[1] > 0 else [a[1], a[2], a[3]],
                                         a,
                                         b,
                                       ),
                               [0, 1, 0],
                               Range(1, Length(_x)),
                             )[3]

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001508

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

live ↗

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

Known value · FindStat St001508

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

live ↗

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

Known value · FindStat St001508

About

Examples	5
Crosswalk	FindStat