A210916 -id:A210916 - cp's OEIS frontend

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

Alois P. Heinz, Dec 04 2007

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

			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, ...

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Alois P. Heinz, Antidiagonals n = 0..140, flattened

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.

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);

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 *)

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).

A218098 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 8.

Original entry on oeis.org

545835, 27733869, 1173919350, 47488375440, 1933688266686, 81009491387682, 3527548086703069, 160415345420268510, 7631859877504516225, 379961855272982538127, 19785139747357478264082, 1076480694153554931849504, 61126131119735946242652270

Offset: 8

Alois P. Heinz, Oct 20 2012

nonn

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Alois P. Heinz, Table of n, a(n) for n = 8..200

Column k=8 of A135313.

t:= proc(k) option remember; `if`(k<0, 0,
      unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
egf:= t(8)(x)-t(7)(x):
a:= n-> n!* coeff(series(egf, x, n+1), x, n):
seq(a(n), n=8..22);

m = 8; t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]] ; egf = t[m][x]-t[m-1][x]; a[n_] := n!*Coefficient[Series[egf, {x, 0, n+1}], x, n]; Table[a[n], {n, m, 22}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

E.g.f.: t_8(x)-t_7(x), with t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) if k>=0 and t_k(x) = 0 else.

a(n) = A210916(n) - A210915(n).

A218099 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 9.

Original entry on oeis.org

7087261, 451238935, 22913136730, 1087116745385, 51075201835515, 2437976801668408, 119752042470064290, 6093096859120003590, 322215964319093498225, 17735784941946000072572, 1016521929886047797022408, 60650840653136697085038930, 3764766650086543657134295955

Offset: 9

Alois P. Heinz, Oct 20 2012

nonn

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Alois P. Heinz, Table of n, a(n) for n = 9..200

Column k=9 of A135313.

t:= proc(k) option remember; `if`(k<0, 0,
      unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
    end:
egf:= t(9)(x)-t(8)(x):
a:= n-> n!* coeff(series(egf, x, n+1), x, n):
seq(a(n), n=9..22);

m = 9; t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]] ; egf = t[m][x]-t[m-1][x]; a[n_] := n!*Coefficient[Series[egf, {x, 0, n+1}], x, n]; Table[a[n], {n, m, 22}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

E.g.f.: t_9(x)-t_8(x), with t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) if k>=0 and t_k(x) = 0 else.

a(n) = A210917(n) - A210916(n).

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A218098 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 8.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A218099 Number of transitive reflexive early confluent binary relations R on n labeled elements with max_{x}(|{y : xRy}|) = 9.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula