A350646 Maximum number of inverses of an element in the full symmetric semigroup T_n.

1, 1, 2, 4, 16, 48, 216, 972, 4374, 24576, 147456, 884736, 5625000, 42187500, 316406250, 2373046875, 19591041024, 176319369216, 1586874322944, 13453731159372, 141264177173406, 1483273860320763, 13843889362993788, 153896443516551168, 1846757322198614016

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

Sequence in context: A003433 A153951 A248748 * A165905 A104354 A153948

Adjacent sequences: A350643 A350644 A350645 * A350647 A350648 A350649

Keyword

nonn

Author

Geoffrey Critzer, Jan 09 2022

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 31 2022

Status

approved