A005345 Number of elements of a free idempotent monoid on n letters.

1, 2, 7, 160, 332381, 2751884514766, 272622932796281408879065987, 3641839910835401567626683593436003894250931310990279692, 848831867913830760986671126293000918118297635181600248839480614255059539078136221019132415247551725144817958905

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn
• easy

Comments

An idempotent monoid satisfies the equation xx=x for any element x.

A squarefree word may be equivalent to a smaller or larger word as a consequence of the idempotent equation.

References

• M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 32.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Morgan Rogers, From free idempotent monoids to free multiplicatively idempotent rigs, arXiv:2408.17440 [math.RA], 2024. See pp. 21, 23.
• Eric Weisstein's World of Mathematics, Monoid.
• Eric Weisstein's World of Mathematics, Free Idempotent Monoid
• Index entries for sequences related to monoids

Crossrefs

A030449(n) = a(n) - 1.

Programs

Mathematica

Array[Sum[Binomial[#, k]* Product[(k - i + 1)^(2^i), {i, k}], {k, 0, #}] &, 10, 0] (* Michael De Vlieger, Sep 05 2024 *)

PARI

{a(n)=sum(k=0, n, binomial(n, k)*prod(i=1, k, (k-i+1)^2^i))} /* Michael Somos, Oct 22 2006 */

Formula

a(n) = Sum_{k=0..n} (C(n, k) Prod_{i=1..k} (k-i+1)^(2^i)).

Binomial transform of A030450. - Michael Somos, Oct 22 2006

Extensions

One more term from Gabriel Cunningham (gcasey(AT)mit.edu), Nov 14 2004