A125625 Number of elements in the semigroup of type K_n.

1, 2, 5, 18, 115, 1710, 83973, 22263378, 64146328635, 5387481983035854, 53332505278384935836485, 448356696524549059043145139274042, 52321110785739610206886887435107004491768788251, 4402157583106925818769478699470667613674438846830415891359277374958, 71336346872409035510345323533390649100576348044074421590685110464047512124710404684631077386973

Offset

0,2

Comments

The semigroup K_3 with 3 generators occurs in convexity theory; K_n is the generic semigroup with n generators.

In the paper of Kudryavtseva and Mazorchuk it is shown that the n-th term of this sequence gives the number of words in the alphabet 1,2,...,n such that between any repetitions of any letter there must occur both a smaller and a bigger letter (in the natural order). For example, the word 2132 is allowed while 3213 is not. [V. Mazorchuk, Aug 24 2011]

Links

Table of n, a(n) for n=0..14.

S. Alsaody, Determining the Elements of a Semigroup, Uppsala, Sweden: Dept. Of Mathematics, Uppsala University, 2007, Report No. 2007:3.

Alessandro D'Andrea and Salvatore Stella, The cardinality of Kiselman's semigroups grows double-exponentially, arXiv:2303.13149 [math.CO], 2023.

L. Forsberg, Effective representations of Hecke-Kiselman monoids of type A, arXiv preprint arXiv:1205.0676 [math.RT], 2012-2017. - From N. J. A. Sloane, Oct 13 2012

C. O. Kiselman, A Semigroup of Operators in Convexity Theory Trans. Am, Math. Soc., 354 (2002), No. 5, pp. 2035-2053.

G. Kudryavtseva and V. Mazorchuk, On Kiselman's semigroup, arXiv:math/0511374 [math.GR], 2005; Preprint Uppsala University 2005, published in: Yokohama Math. J. 55 (2009), no.1, 21-46.

M. Selin, Source code (C++) for algorithm.

Prog

(Magma) /* program for a(6) */ F<a, b, c, d, e, f> := FreeMonoid(6); Q<a, b, c, d, e, f> := quo< F | a^2 = a, b^2 = b, c^2 = c, d^2 = d, e^2 = e, f^2 = f, a*b*a = b*a*b = a*b, a*c*a = c*a*c = a*c, a*d*a = d*a*d = a*d, a*e*a = e*a*e = a*e, a*f*a = f*a*f = a*f, b*c*b = c*b*c = b*c, b*d*b = d*b*d = b*d, b*e*b = e*b*e = b*e, b*f*b = f*b*f = b*f, c*d*c = d*c*d = c*d, c*e*c = e*c*e = c*e, c*f*c = f*c*f = c*f, d*e*d = e*d*e = d*e, d*f*d = f*d*f = d*f, e*f*e = f*e*f = e*f >; M<a, b, c, d, e, f> := RWSMonoid(Q); Order(M); /* Klaus Brockhaus, Mar 02 2007 */

Crossrefs

Sequence in context: A005805 A058338 A006896 * A281532 A097584 A197855

Adjacent sequences: A125622 A125623 A125624 * A125626 A125627 A125628

Keyword

nonn

Author

Seidon Alsaody (Seidon.Alsaody.5527(AT)student.uu.se), Jan 27 2007

Extensions

a(6) from Klaus Brockhaus, Mar 02 2007

More terms from M. Selin (mxrten(AT)kth.se), Jan 16 2008, Jan 25 2008

Status

approved