SetValuedTableauNaturalDescents

The natural descents of the set-valued two-row tableau the path corresponds to, read off the step pairs by FindStat's rules.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001955via DyckPath✓↗

SetValuedTableauNaturalDescents(path)The natural descents of the set-valued two-row tableau the path corresponds to, read off the step pairs by FindStat's rules.

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 St001955. The rules go through bicolored Motzkin paths; they are stated on the path itself and implemented as given.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (j: integer) => Length(Filter(
                                                               Range(
                                                                      1,
                                                                      Length(_x) /
                                                                      2 + 1,
                                                                    ),
                                                               Function(
                                                                         k - 2 >=
                                                                         j && (_x[2k - 3] if 2k - 3 >=
                                                                                            1 &&
                                                                                            2k - 3 <=
                                                                                            Length(_x) else 2) +
                                                                             (_x[2k - 2] if 2k - 2 >=
                                                                                            1 &&
                                                                                            2k - 2 <=
                                                                                            Length(_x) else 2) ==
                                                                             1 ||
                                                                         k - 1 >=
                                                                         j && (_x[2k - 2] if 2k - 2 >=
                                                                                            1 &&
                                                                                            2k - 2 <=
                                                                                            Length(_x) else 2) ==
                                                                             0 &&
                                                                             (_x[2k - 1] if 2k - 1 >=
                                                                                            1 &&
                                                                                            2k - 1 <=
                                                                                            Length(_x) else 2) ==
                                                                             1 &&
                                                                             (_x[2k - 0] if 2k - 0 >=
                                                                                            1 &&
                                                                                            2k - 0 <=
                                                                                            Length(_x) else 2) ==
                                                                             1 || k ==
                                                                             j &&
                                                                             (_x[2k - 1] if 2k - 1 >=
                                                                                            1 &&
                                                                                            2k - 1 <=
                                                                                            Length(_x) else 2) ==
                                                                             0 &&
                                                                             (_x[2k - 0] if 2k - 0 >=
                                                                                            1 &&
                                                                                            2k - 0 <=
                                                                                            Length(_x) else 2) ==
                                                                             0 ||
                                                                         k < j &&
                                                                         (_x[2k -
                                                                             1] if 2k - 1 >=
                                                                                  1 &&
                                                                                  2k - 1 <=
                                                                                  Length(_x) else 2) ==
                                                                         0 && (_x[2k - 0] if 2k - 0 >=
                                                                                            1 &&
                                                                                            2k - 0 <=
                                                                                            Length(_x) else 2) ==
                                                                             1,
                                                                         k,
                                                                       ),
                                                             )),
                                Min(Filter(
                                            Range(1, Length(_x) / 2),
                                            Function(_x[2i] == 0, i),
                                          )),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{SetValuedTableauNaturalDescents}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$2$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001955

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

live ↗

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

Known value · FindStat St001955

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

live ↗

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

Known value · FindStat St001955

About

Examples	5
Crosswalk	FindStat