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%I #24 Mar 28 2023 17:37:06
%S 1,2,5,18,115,1710,83973,22263378,64146328635,5387481983035854,
%T 53332505278384935836485,448356696524549059043145139274042,
%U 52321110785739610206886887435107004491768788251,4402157583106925818769478699470667613674438846830415891359277374958,71336346872409035510345323533390649100576348044074421590685110464047512124710404684631077386973
%N Number of elements in the semigroup of type K_n.
%C The semigroup K_3 with 3 generators occurs in convexity theory; K_n is the generic semigroup with n generators.
%C 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]
%H S. Alsaody, <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=4135c9ec39a4c0f9b76069fe8268f0e0de01d09e">Determining the Elements of a Semigroup</a>, Uppsala, Sweden: Dept. Of Mathematics, Uppsala University, 2007, Report No. 2007:3.
%H Alessandro D'Andrea and Salvatore Stella, <a href="https://arxiv.org/abs/2303.13149">The cardinality of Kiselman's semigroups grows double-exponentially</a>, arXiv:2303.13149 [math.CO], 2023.
%H L. Forsberg, <a href="http://arxiv.org/abs/1205.0676">Effective representations of Hecke-Kiselman monoids of type A</a>, arXiv preprint arXiv:1205.0676 [math.RT], 2012-2017. - From _N. J. A. Sloane_, Oct 13 2012
%H C. O. Kiselman, <a href="https://doi.org/10.1090/S0002-9947-02-02915-X">A Semigroup of Operators in Convexity Theory</a> Trans. Am, Math. Soc., 354 (2002), No. 5, pp. 2035-2053.
%H G. Kudryavtseva and V. Mazorchuk, <a href="https://arxiv.org/abs/math/0511374">On Kiselman's semigroup</a>, arXiv:math/0511374 [math.GR], 2005; Preprint Uppsala University 2005, published in: Yokohama Math. J. 55 (2009), no.1, 21-46.
%H M. Selin, <a href="http://www.f.kth.se/~mxrten/kiselman_semigroup.cpp">Source code</a> (C++) for algorithm.
%o (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 */
%K nonn
%O 0,2
%A Seidon Alsaody (Seidon.Alsaody.5527(AT)student.uu.se), Jan 27 2007
%E a(6) from _Klaus Brockhaus_, Mar 02 2007
%E More terms from M. Selin (mxrten(AT)kth.se), Jan 16 2008, Jan 25 2008
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