A370208 Triangular array read by rows. T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.
1, 1, 1, 3, 6, 13, 39, 87, 348, 24, 841, 4205, 480, 11643, 69858, 9420, 240, 227893, 1595251, 206640, 9240, 6285807, 50286456, 5389552, 299040, 3360, 243593041, 2192337369, 172041408, 9848160, 211680
Offset: 0
Examples
Triangle begins
1;
1, 1;
3, 6;
13, 39;
87, 348, 24;
841, 4205, 480;
11643, 69858, 9420, 240;
227893, 1595251, 206640, 9240;
...
Links
- David Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research, National Bureau of Standards, Vol 67B No. 4 Oct-Dec 1963.
Crossrefs
Programs
-
Mathematica
nn = 9; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}]; c[x_] := Log[A[x]] - x; Map[Select[#, # > 0 &] &, Range[0, nn]! CoefficientList[ Series[2 (Exp[ y x D[c[ x], x]/2] - 1) Exp[c[x]] Exp[ x] + Exp[c[ x]] (y x Exp[ x] + Exp[ x]), {x, 0, nn}], {x, y}]]
Formula
E.g.f.: 2(exp(y*x*c'(x)/2)-1)*exp(c(x))*exp(x) + exp(c(x))*(y*x*exp(x) + exp(x)) where c(x) is the e.g.f. for A002031.