OFFSET
0,3
COMMENTS
Let f,g be in T_n, the semigroup of all functions from [n] into [n]. Then f,g are an inverse pair if fgf=f and gfg=g. Let V(f) = {g in T_n:f and g are an inverse pair}. Then |V(f)| = m_1*m_2*...*m_k*k^(n-k) where image(f)={a_1,a_2,...,a_k} and m_i=|{x in [n]:f(x) = a_i}|. Then a(n) = max{|V(f):f in T_n|}.
REFERENCES
O. Ganyuskin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 25.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..80
MATHEMATICA
f[p_] := Apply[Times, p]*Length[p]^(Total[p] - Length[p]); Table[Max[Map[f, IntegerPartitions[n]]], {n, 1, 22}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jan 09 2022
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jan 31 2022
STATUS
approved