SetValuedTableauComajorIndex

The comajor index of the set-valued two-row tableau the path corresponds to: the sum, over its natural descents k, of the semilength plus one less k.

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SetValuedTableauComajorIndex(path)The comajor index of the set-valued two-row tableau the path corresponds to: the sum, over its natural descents k, of the semilength plus one less k.

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 St001956; the natural descents are those of SetValuedTableauNaturalDescents.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply((j: integer) => Sum(Map(
                                                        Function(Length(_x) / 2 +
                                                                 1 - k, k),
                                                        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$\mathrm{SetValuedTableauComajorIndex}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$2$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001956

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

live ↗

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

Known value · FindStat St001956

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

live ↗

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

Known value · FindStat St001956

About

Examples	5
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