A132431 For n>0, let B_n be the subsemigroup of the full transformation monoid on the n-set [n] generated by the following functions: Let x be a certain element in [n]. Now the generators of B are those functions which map either x to any distinct element y in [n] leaving all the other elements fixed, or y to x leaving all the other elements fixed. Then a(n) = number of elements in B_n.

0, 2, 9, 88, 1385, 24336, 466753, 9906688, 233522577, 6093136000, 174912502721, 5487091383456, 186891076515481, 6870622015481056, 271195480556337345, 11440127985767481856, 513639921634424850977, 24455974520989478444544, 1230835712617872016215265

Offset: 1

Author

Simon Bogner (sisibogn(AT)stud.informatik.uni-erlangen.de), Nov 20 2007

Keywords

• nice
• nonn

Comments

Let b(n)=n^n be the cardinality of the full transformation monoid. The sequence of quotients a(n)/b(n) converges to 1-1/e.

References

• S. Bogner, Eine Praesentation der Halbgruppe der singularen zyklisch-monotonen Abbildungen UND eine von Idempotenten erzeugte Unterhalbgruppe von T_n (Studienarbeit in Informatik, Advisor: Klaus Leeb), Friedrich-Alexander-Universitaet Erlangen-Nuernberg, 2007.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A000312, A060226.

Programs

Haskell

a132431 n = a060226 n - a062119 n + a002378 (n - 1)
-- Reinhard Zumkeller, Aug 27 2012

Mathematica

Join[{0},Table[n^n-n (n-1)^(n-1)-(n-1)n!+n(n-1),{n,2,20}]] (* Harvey P. Dale, Jun 07 2018 *)

Formula

a(n) = n^n - n*(n-1)^(n-1) - (n-1)*n! + n*(n-1).

a(n) = n*(n-1) + Sum_{k=1..n-2} k*Stirling2(n-1,k)*k!*C(n,k).

a(n) = A060226(n) - A062119(n) + A002378(n-1). - Reinhard Zumkeller, Aug 27 2012