DistinctSubsequenceCount

The distinct subsequences, the empty one included.

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DistinctSubsequenceCount(word)The distinct subsequences, the empty one included.

Typed signatures

Type	Package	Applies
(binary_word) -> number	combinatorics	always

Domain: Binary word statistics

Details

• Defined over BinaryWord as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a BinaryWord 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 St000518. With d(j) the count for the first j letters, d(j) = 2 d(j - 1) less d(p - 1), where p is the latest earlier position holding the same letter, if any.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

1 if Length(_x) < 1 else Apply((a_init: list<integer>) => Fold(
                                                                (
                                                                  a: list<integer>,
                                                                  b,
                                                                ) => Apply(
                                                                           (a_next: list<integer>) => a_next,
                                                                           Join(
                                                                                 a,
                                                                                 [2 * a[Length(a)] -
                                                                                  a[Max(Join(
                                                                                              [0],
                                                                                              Map(
                                                                                                   Function(p if _x[p] == _x[b] else 0, p),
                                                                                                   Range(1, b - 1, 1),
                                                                                                 ),
                                                                                            )) +
                                                                                    1]],
                                                                               ),
                                                                         ),
                                                                a_init,
                                                                Range(
                                                                       1,
                                                                       Length(_x),
                                                                     ),
                                                              ), [0, 1])[Length(_x) +
                                                                        2]

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{DistinctSubsequenceCount}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$78$

Properties #

FindStat's value for 0

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([0]), \text{DistinctSubsequenceCount})$Out$2$

Known value · FindStat St000518

FindStat's value for 1

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([1]), \text{DistinctSubsequenceCount})$Out$2$

Known value · FindStat St000518

FindStat's value for 00

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([0, 0]), \text{DistinctSubsequenceCount})$Out$3$

Known value · FindStat St000518

About

Examples	4
Crosswalk	FindStat