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A242432 Length of longest chain of nonempty proper subsemigroups of the monoid of partial injective orientation-preserving functions of a chain with n elements.

%I A242432 #21 Feb 19 2019 04:06:53
%S A242432 1,6,24,92,363,1483,6191,26077,109987,462900,1941613,8115138,33805905,
%T A242432 140413073,581694265,2404314784,9917782935,40837958578,167889571658,
%U A242432 689231516287,2825851058202,11572537702747,47342211484912,193485587828057,790066214186999,3223470297388819,13141840760544209,53540833421980514
%N A242432 Length of longest chain of nonempty proper subsemigroups of the monoid of partial injective orientation-preserving functions of a chain with n elements.
%H A242432 James Mitchell, <a href="/A242432/b242432.txt">Table of n, a(n) for n = 1..100</a>
%H A242432 P. J. Cameron, M. Gadouleau, J. D. Mitchell, Y. Peresse, <a href="http://arxiv.org/abs/1501.06394">Chains of subsemigroups</a>, arXiv preprint arXiv:1501.06394 [math.GR], 2015.
%t A242432 b[n_] := If[n < 1, 0, PrimeOmega[n]];
%t A242432 a[n_] := -2 - n + Sum[Binomial[n, i]*(b[i] + (Binomial[n, i] - 1)*i/2 + 2), {i, 0, n}];
%t A242432 Array[a, 28] (* _Jean-François Alcover_, Feb 19 2019, from PARI *)
%o A242432 (PARI) b(n)=if(n<1, 0, bigomega(n)) /* A001222 */
%o A242432 a(n)=-2-n+sum(i=0, n, binomial(n,i)*(b(i)+(binomial(n,i)-1)*i/2+2))
%Y A242432 Cf. A001222, A227914, A242428, A242429.
%K A242432 nonn
%O A242432 1,2
%A A242432 _James Mitchell_, May 14 2014