A210916 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 8 for all x.

1, 1, 4, 26, 243, 2992, 45906, 845287, 18182926, 440710385, 11876274391, 351546957499, 11330575607067, 394862762014644, 14792903605828298, 592835563146850723, 25306351970600498930, 1146305330627242918543, 54914971513967144548105, 2773947252964889935144249

Offset: 0

Author

Alois P. Heinz, Mar 29 2012

Keywords

• nonn

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

References

• A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=8 of A135302.

Programs

Maple

t:= proc(k) option remember;
      `if`(k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
gf:= t(8)(x):
a:= n-> n!* coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

Mathematica

t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]]; gf = t[8][x]; a[n_] := n!*SeriesCoefficient[gf, {x, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2014, translated from Maple *)

Formula

E.g.f.: t_8(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) for k>=0 and t_k(x) = 0 otherwise.