A242428 Length of longest chain of nonempty proper subsemigroups of the dual symmetric inverse monoid.

0, 2, 17, 180, 3298, 88431, 3064050, 130905678, 6732227475, 409094032964, 28917250469178, 2346562701385648, 216180120430479731, 22397392442055209003, 2588479398843886168171, 331352273262513644199134, 46692196905193286953380160, 7203294536351261350956567853, 1210694223244114528129261255186

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. A007238, A227914, A023997.

Programs

Mathematica

a[n_] := Sum[StirlingS2[n, i] (i! (StirlingS2[n, i] - 1)/2 - DigitCount[i, 2, 1] + Ceiling[3 i/2] + 1), {i, 1, n}] - n - 1;
Array[a, 19] (* Jean-François Alcover, Dec 12 2018, from PARI *)

PARI

b(n)=if(n<1, 0, b(n\2)+n%2) /* A000120 */
a(n)=-n-1+sum(i=1, n, stirling(n,i,flag=2)*(ceil(3*i/2)-b(i)+1+(stirling(n,i,flag=2)-1)*i!/2))