A210913 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 5 for all x.

1, 1, 4, 26, 243, 2992, 41223, 660220, 11979669, 243048992, 5448497434, 133595966164, 3555887814602, 102064563003898, 3141580135645330, 103198691666336823, 3602725068242695657, 133174089439019869924, 5195463138498447345478, 213295995976349091757666

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=5 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(5)(x):
a:= n-> n!*coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

Mathematica

t[0, ] = 1; t[k, x_] := t[k, x] = Exp[Sum[x^m/m!*t[k - m, x], {m, 1, k}]]; a[0, 0] = 1; a[, 0] = 0; a[n, k_] := SeriesCoefficient[t[k, x], {x, 0, n}]*n!; Table[a[n, 5], {n, 0, 30} ] (* Jean-François Alcover, Feb 04 2014, after A135302 and Alois P. Heinz *)

Formula

E.g.f.: t_5(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.