%I A135313
%S A135313 1,0,1,0,1,3,0,1,12,13,0,1,61,106,75,0,1,310,1105,1035,541,0,1,1821,
%T A135313 12075,16025,11301,4683,0,1,11592,141533,267715,239379,137774,47293,0,
               1,
%U A135313 80963,1812216,4798983,5287506,3794378,1863044,545835,0,1,608832
%N A135313 Triangle of numbers T(n,k) (n>=0, n>=k>=0) of transitive reflexive early 
               confluent binary relations R on n labeled elements where k=max_{x}(|{y 
               : xRy}|), read by rows. Early confluency means that (xRy AND xRz) 
               implies (yRz OR zRy) for all x, y, z.
%C A135313 Triangle begins : 1 0, 1 0, 1, 3 0, 1, 12, 13 0, 1, 61, 106, 75 0, 1, 
               310, 1105, 1035, 541 ... Diagonal is sequence A000670.
%D A135313 A. P. Heinz (1990). Analyse der Grenzen und Moeglichkeiten schneller 
               Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universitaet Freiburg, 
               Freiburg i. Br., Germany.
%H A135313 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Dec 05 2007, <a href="b135313.txt">
               Table of n, a(n) for n = 0..209</a>
%F A135313 T(n,0)=A135302(n,0), T(n,k)=A135302(n,k)-A135302(n,k-1) for k>0; e.g.f. 
               of column k=0: tt_0(x)=1, e.g.f. of column k>0: tt_k(x)=t_k(x)-t_{k-1}(x), 
               where t_k(x)=exp(sum_{m=1..k}(x^m/m!*t_{k-m}(x))) if k>=0 and t_k(x)=0 
               else.
%e A135313 T(3,3)=13 because there are 13 relations of the given kind for 3 elements:
%e A135313 1R2, 2R1, 1R3, 3R1, 2R3, 3R2;
%e A135313 1R2, 1R3, 2R3, 3R2;
%e A135313 2R1, 2R3, 1R3, 3R1;
%e A135313 3R1, 3R2, 1R2, 2R1;
%e A135313 2R1, 3R1, 2R3, 3R2;
%e A135313 1R2, 3R2, 1R3, 3R1;
%e A135313 1R3, 2R3, 1R2, 2R1;
%e A135313 1R2, 2R3, 1R3;
%e A135313 1R3, 3R2, 1R2;
%e A135313 2R1, 1R3, 2R3;
%e A135313 2R3, 3R1, 2R1;
%e A135313 3R1, 1R2, 3R2;
%e A135313 3R2, 2R1, 3R1;
%e A135313 (the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity)
%p A135313 t := proc(k) option remember; if k<0 then 0 else unapply(exp(sum('x^m/
               m!*t(k-m)(x)', 'm'=1..k)), x) fi; end; tt := proc(k) option remember; 
               unapply((t(k)-t(k-1))(x), x); end; T := proc(n,k) option remember; 
               coeff(series(tt(k)(x), x=0, n+1), x, n)*n!; end; seq(seq(T(n,k), 
               k=0..n), n=0..12);
%Y A135313 Cf. A135302, A000670.
%Y A135313 Row sums are in A052880. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Jun 01 2009]
%Y A135313 Sequence in context: A112906 A137375 A145881 this_sequence A022695 A067169 
               A011339
%Y A135313 Adjacent sequences: A135310 A135311 A135312 this_sequence A135314 A135315 
               A135316
%K A135313 nonn,tabl
%O A135313 0,6
%A A135313 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Dec 05 2007