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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. 20
 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, 168, 26, 4, 1, 1, 0, 1, 1822, 1416, 243, 26, 4, 1, 1, 0, 1, 11593, 13897, 2451, 243, 26, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z. REFERENCES A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA 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)). A(n,k) = Sum_{i=0..k} A135313(n,i). EXAMPLE Table A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 0, 1, 1, 1, 1, 1, ... 0, 1, 4, 4, 4, 4, ... 0, 1, 13, 26, 26, 26, ... 0, 1, 62, 168, 243, 243, ... 0, 1, 311, 1416, 2451, 2992, ... MAPLE t:= proc(k) option remember; `if`(k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x)) end: A:= proc(n, k) option remember; coeff(series(t(k)(x), x, n+1), x, n) *n! end: seq(seq(A(d-i, i), i=0..d), d=0..15); MATHEMATICA 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 *) CROSSREFS Columns k=0-10 give: A000007, A000012, A135312, A210911, A210912, A210913, A210914, A210915, A210916, A210917, A210918. Main diagonal gives A052880. A(n,n)-A(n,n-1) gives A000670. Cf. A135313. Sequence in context: A085639 A158972 A278987 * A128760 A057884 A329637 Adjacent sequences: A135299 A135300 A135301 * A135303 A135304 A135305 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 04 2007 STATUS approved

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Last modified April 1 03:25 EDT 2023. Contains 361673 sequences. (Running on oeis4.)