KBracelets

Words of length $n$ over a $k$-letter alphabet up to rotation and reflection — bracelets over $k$ colours, a lazy indexed family.

SymPybracelets↗

KBracelets(n, k)the bracelets of $n$ beads, each one of $k$ colours.

Typed signatures

Type	Package	Applies
(integer<0..>, integer<0..>) -> indexed_collection<list<integer>>	combinatorics	always

Domain: Collections

Details

• A lazy indexed collection; generalises BinaryBracelets from $k=2$ to any alphabet size, by the same Burnside sum over $D_n$.
• Each element is the lexicographically-least word in its rotation-and-reflection orbit, over letters $0,…,k-1$.
• $At$ unranks over these canonical words in ascending lexicographic order.

Enumeration

Examples

1 more kept as data, checked against the oracles

Basic #

$Count(KBracelets(4, 3)) = 21$: bracelets of length 4 over a 3-letter alphabet

live ↗

In$\mathrm{Count}(\mathrm{KBracelets}(4, 3))$Out$21$

Scope #

Exact past $2^{53}$: 3-color bracelets of 41 beads

live ↗

In$\mathrm{Count}(\mathrm{KBracelets}(41, 3))$Out$444\,792\,643\,976\,161\,803$

Known value · OEIS A027671 b-file

Properties #

$KBracelets(n, 2)$ agrees with BinaryBracelets$(n)$ — same Burnside sum over $D_n$

live ↗

In$\mathrm{Count}(\mathrm{KBracelets}(5, 2))=\mathrm{Count}(\mathrm{BinaryBracelets}(5))$Out$\mathrm{True}$

The 3-color bracelets of $n$ beads, $n = 0 \dots 17$

live ↗

In$\mathrm{Map}(n↦\mathrm{Count}(\mathrm{KBracelets}(n, 3)), 0..1..17)$Out$\bigl\lbrack1, 3, 6, 10, 21, 39, 92, 198, 498, 1\,219, 3\,210, 8\,418, 22\,913, 62\,415, 173\,088, 481\,598, 1\,351\,983, 3\,808\,083\bigr\rbrack$

Known value · OEIS A027671

About

Examples	5
Crosswalk	SymPy

See also: BinaryBracelets, Totient, Count, At