Coarea

The complement of the area within the staircase: binomial(n, 2) less the area.

Coarea(path)The complement of the area within the staircase: binomial(n, 2) less the area.

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.
• Zero for the highest path, UU...DD, whose area is binomial(n, 2).

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Length(_x) / 2 * (Length(_x) / 2 - 1) / 2 -
                         (0 if Length(_x) < 1 else (Sum(Map(
                                                             Function(Sum(Map(
                                                                               Function(2 * _x[k] -
                                                                                        1, k),
                                                                               Range(1, i),
                                                                             )), i),
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                           )) - Length(_x) / 2) /
                                                   2)

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

About

Examples	2