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

1, 6, 24, 92, 363, 1483, 6191, 26077, 109987, 462900, 1941613, 8115138, 33805905, 140413073, 581694265, 2404314784, 9917782935, 40837958578, 167889571658, 689231516287, 2825851058202, 11572537702747, 47342211484912, 193485587828057, 790066214186999, 3223470297388819, 13141840760544209, 53540833421980514

Offset: 1

Author

James Mitchell, May 14 2014

Keywords

• nonn

Links

• James Mitchell, Table of n, a(n) for n = 1..100
• P. J. Cameron, M. Gadouleau, J. D. Mitchell, Y. Peresse, Chains of subsemigroups, arXiv preprint arXiv:1501.06394 [math.GR], 2015.

Crossrefs

Cf. A001222, A227914, A242428, A242429.

Programs

Mathematica

b[n_] := If[n < 1, 0, PrimeOmega[n]];
a[n_] := -2 - n + Sum[Binomial[n, i]*(b[i] + (Binomial[n, i] - 1)*i/2 + 2), {i, 0, n}];
Array[a, 28] (* Jean-François Alcover, Feb 19 2019, from PARI *)

PARI

b(n)=if(n<1, 0, bigomega(n)) /* A001222 */
a(n)=-2-n+sum(i=0, n, binomial(n,i)*(b(i)+(binomial(n,i)-1)*i/2+2))