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%I #44 Dec 29 2021 16:37:08
%S 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 168,26,4,1,1,0,1,1822,1416,243,26,4,1,1,0,1,11593,13897,2451,243,26,
%U 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 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 R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
%D A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.
%H Alois P. Heinz, <a href="/A135302/b135302.txt">Antidiagonals n = 0..140, flattened</a>
%F 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 A(n,k) = Sum_{i=0..k} A135313(n,i).
%e Table A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, ...
%e 0, 1, 4, 4, 4, 4, ...
%e 0, 1, 13, 26, 26, 26, ...
%e 0, 1, 62, 168, 243, 243, ...
%e 0, 1, 311, 1416, 2451, 2992, ...
%p t:= proc(k) option remember; `if`(k<0, 0,
%p unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
%p end:
%p A:= proc(n, k) option remember;
%p coeff(series(t(k)(x), x, n+1), x, n) *n!
%p end:
%p seq(seq(A(d-i, i), i=0..d), d=0..15);
%t 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 Columns k=0-10 give: A000007, A000012, A135312, A210911, A210912, A210913, A210914, A210915, A210916, A210917, A210918.
%Y Main diagonal gives A052880.
%Y A(n,n)-A(n,n-1) gives A000670.
%Y Cf. A135313.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Dec 04 2007
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