0,3

The property (xy)z = xy for a binary operation is the action condition for a left projection semigroup (product xy = x) acting as a semigroup on the right on itself as a set. Constants have this property, and the left projection, but not the right projection. All of the right translations must be idempotent functions with the same set of fixed points.

This sequence is R(n, n) where R(m, n) = Sum_{r=1..n} binomial(n,r)*r^((n-r)*m) is the number of right actions of the left projection semigroup of m elements acting on n elements.

The binary operation property that is the action condition for left actions of the left projection on itself as semigroup on set is x(yz) = xz, counted by A279644.

Paolo Xausa, Table of n, a(n) for n = 0..36

a(n) = Sum_{r=0..n} binomial(n,r)*r^((n-r)*n).

A390603[n_] := If[n == 0, 1, Sum[Binomial[n, r]*r^((n - r)*n), {r, n}]];

Array[A390603, 15, 0] (* Paolo Xausa, Nov 22 2025 *)

(PARI) a(n) = sum(r=0, n, binomial(n, r)*r^((n-r)*n)); \\ Michel Marcus, Nov 12 2025

Cf. A002489, A279644, A390164 (unlabeled case).

Sequence in context: A196735 A208438 A122891 * A119156 A175979 A096621

Adjacent sequences: A390600 A390601 A390602 * A390604 A390605 A390606

David Pasino, Nov 12 2025

approved