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A135302
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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.
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20
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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
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OFFSET
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0,13
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COMMENTS
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R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
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REFERENCES
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A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..140, flattened
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FORMULA
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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).
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Dec 04 2007
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STATUS
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approved
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