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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.

%I A135313 #44 Nov 20 2021 10:52:22
%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,
%U A135313 1,80963,1812216,4798983,5287506,3794378,1863044,545835,0,1,608832,25188019,92374107,124878033,105494886,64432638,27733869,7087261
%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.
%C A135313 R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
%C A135313 Conjecture: For fixed k>=0, A135313(n+k,n) ~ n! * n^(2*k)  / (2^(k+1) * k! * log(2)^(n+k+1)). - _Vaclav Kotesovec_, Nov 20 2021
%D A135313 A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.
%H A135313 Alois P. Heinz, <a href="/A135313/b135313.txt">Rows n = 0..140, flattened</a>
%F A135313 T(n,0) = A135302(n,0), T(n,k) = A135302(n,k) - A135302(n,k-1) for k>0.
%F A135313 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:  (1) 1R2, 2R1, 1R3, 3R1, 2R3, 3R2;  (2) 1R2, 1R3, 2R3, 3R2;  (3) 2R1, 2R3, 1R3, 3R1;  (4) 3R1, 3R2, 1R2, 2R1;  (5) 2R1, 3R1, 2R3, 3R2; (6) 1R2, 3R2, 1R3, 3R1;  (7) 1R3, 2R3, 1R2, 2R1;  (8) 1R2, 2R3, 1R3;  (9) 1R3, 3R2, 1R2;  (10) 2R1, 1R3, 2R3;  (11) 2R3, 3R1, 2R1;  (12) 3R1, 1R2, 3R2;  (13) 3R2, 2R1, 3R1; (the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity).
%e A135313 Triangle T(n,k) begins:
%e A135313   1;
%e A135313   0,  1;
%e A135313   0,  1,    3;
%e A135313   0,  1,   12,    13;
%e A135313   0,  1,   61,   106,    75;
%e A135313   0,  1,  310,  1105,  1035,   541;
%e A135313   0,  1, 1821, 12075, 16025, 11301, 4683;
%e A135313   ...
%p A135313 t:= proc(k) option remember; `if`(k<0, 0,
%p A135313       unapply(exp(add(x^m/m!*t(k-m)(x), m=1..k)), x))
%p A135313     end:
%p A135313 tt:= proc(k) option remember;
%p A135313        unapply((t(k)-t(k-1))(x), x)
%p A135313      end:
%p A135313 T:= proc(n, k) option remember;
%p A135313       coeff(series(tt(k)(x), x, n+1), x, n)*n!
%p A135313     end:
%p A135313 seq(seq(T(n, k), k=0..n), n=0..12);
%t A135313 f[0, _] = 1; f[k_, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k - m, x], {m, 1, k}]]; (* a = A135302 *) a[0, 0] = 1; a[_, 0] = 0; a[n_, k_] := SeriesCoefficient[f[k, x], {x, 0, n}]*n!; t[n_, 0] := a[n, 0]; t[n_, k_] := a[n, k] - a[n, k-1]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 06 2013, after A135302 *)
%Y A135313 Columns k=0-10 give: A000007, A057427, A218092, A218093, A218094, A218095, A218096, A218097, A218098, A218099, A218091.
%Y A135313 Main diagonal and lower diagonals give: A000670, A218111, A218112, A218103, A218104, A218105, A218106, A218107, A218108, A218109, A218110.
%Y A135313 Row sums are in A052880.
%Y A135313 T(2n,n) gives A261238.
%Y A135313 Cf. A135302.
%K A135313 nonn,tabl
%O A135313 0,6
%A A135313 _Alois P. Heinz_, Dec 05 2007