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.

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

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: A158972 A372173 A278987 * A364026 A128760 A057884

Adjacent sequences: A135299 A135300 A135301 * A135303 A135304 A135305

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 04 2007

Status

approved