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A135302 Square array of numbers A(n,k) (n>=0, k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= k for all x, read by antidiagonals.

%I A135302 #44 Dec 29 2021 16:37:08
%S A135302 1,0,1,0,1,1,0,1,1,1,0,1,4,1,1,0,1,13,4,1,1,0,1,62,26,4,1,1,0,1,311,
%T A135302 168,26,4,1,1,0,1,1822,1416,243,26,4,1,1,0,1,11593,13897,2451,243,26,
%U A135302 4,1,1,0,1,80964,153126,29922,2992,243,26,4,1,1,0,1,608833,1893180,420841,41223,2992,243,26,4,1,1
%N A135302 Square array of numbers A(n,k) (n>=0, k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= k for all x, read by antidiagonals.
%C A135302 R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
%D A135302 A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.
%H A135302 Alois P. Heinz, <a href="/A135302/b135302.txt">Antidiagonals n = 0..140, flattened</a>
%F A135302 E.g.f. of column k=0: t_0(x) = 1;  e.g.f. of column k>0: t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)).
%F A135302 A(n,k) = Sum_{i=0..k} A135313(n,i).
%e A135302 Table A(n,k) begins:
%e A135302   1, 1,   1,    1,    1,    1, ...
%e A135302   0, 1,   1,    1,    1,    1, ...
%e A135302   0, 1,   4,    4,    4,    4, ...
%e A135302   0, 1,  13,   26,   26,   26, ...
%e A135302   0, 1,  62,  168,  243,  243, ...
%e A135302   0, 1, 311, 1416, 2451, 2992, ...
%p A135302 t:= proc(k) option remember; `if`(k<0, 0,
%p A135302        unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
%p A135302     end:
%p A135302 A:= proc(n, k) option remember;
%p A135302       coeff(series(t(k)(x), x, n+1), x, n) *n!
%p A135302     end:
%p A135302 seq(seq(A(d-i, i), i=0..d), d=0..15);
%t A135302 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-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 06 2013, after Maple *)
%Y A135302 Columns k=0-10 give: A000007, A000012, A135312, A210911, A210912, A210913, A210914, A210915, A210916, A210917, A210918.
%Y A135302 Main diagonal gives A052880.
%Y A135302 A(n,n)-A(n,n-1) gives A000670.
%Y A135302 Cf. A135313.
%K A135302 nonn,tabl
%O A135302 0,13
%A A135302 _Alois P. Heinz_, Dec 04 2007