A135458 Number of transitive reflexive binary relations R on n labeled elements where max_{x}(|{y : xRy}|)=3.

0, 0, 0, 16, 148, 1805, 23700, 351239, 5919312, 112984855, 2429692570, 58481205365, 1564981962720, 46269631044377, 1502736397861062, 53336395962363115, 2059205384354896000, 86117408372404734527, 3886421055028996900050, 188615545123265662965785

Offset: 0

Author

Alois P. Heinz, Dec 15 2007

Keywords

• nonn

Examples

			a(3)=16 because there are 16 relations of the given kind for 3 elements:
1R2, 2R1, 1R3, 3R1, 2R3, 3R2;
1R2, 1R3, 2R3, 3R2;
2R1, 2R3, 1R3, 3R1;
3R1, 3R2, 1R2, 2R1;
1R2, 2R1, 1R3, 2R3;
1R3, 3R1, 1R2, 3R2;
2R3, 3R2, 2R1, 3R1;
1R2, 2R3, 1R3;
1R3, 3R2, 1R2;
2R1, 1R3, 2R3;
2R3, 3R1, 2R1;
3R1, 1R2, 3R2;
3R2, 2R1, 3R1;
1R2, 1R3;
2R1, 2R3;
3R1, 3R2;
(the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity)
		

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, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A135429, A135312, A025035, A008277, A006882, A007318, A000248, A000142.

Programs

Maple

A025035:= proc(n) option remember; (3*n)! /n! /(6^n); end:
z:= proc(n) option remember; add(binomial(n, k+k) *doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2)); end:
r:= proc(n) option remember; n! * add(add(add(add(Stirling2(e, d) *a^(d+i) *(a*(a+1)/2)^(n-i-i-e-d-a) /a! /(n-i-i-e-d-a)! /i! /e! /(2^i), a=0..(n-i-i-e-d)), d=0..min(e, n-i-i-e)), e=0..(n-i-i)), i=0..floor(n/2)) end:
A135429:= proc(n) option remember; n! *add(add(A025035(i) *z(j) *r(n-3*i-j) /(3*i)! /j! /(n-3*i-j)!, j=0..(n-3*i)), i=0..floor(n/3)) end:
A000248:= proc(n) add(binomial(n, i)*(n-i)^i, i=0..n) end:
A135312:= proc(n) option remember; add(binomial(n, i+i)*doublefactorial(i+i-1)*A000248(n-i-i), i=0..floor(n/2)) end:
a:= proc(n) A135429(n)-A135312(n) end:
seq(a(i), i=0..30);

Mathematica

Unprotect[Power]; 0^0 = 1; A025035[n_] := A025035[n] = (3n)!/n!/6^n; z[n_] := z[n] = Sum[Binomial[n, k+k]*(k+k-1)!!*k^(n-k-k), {k, 0, Floor[n/2]}]; r[n_] := r[n] = n!*Sum[Sum[Sum[Sum[StirlingS2[e, d]*a^(d+i)*(a*(a+1)/2)^(n-i-i-e-d-a)/a!/(n-i-i-e-d-a)!/i!/e!/2^i, {a, 0, n-i-i-e-d}], {d, 0, Min[e, n-i-i-e]}], {e, 0, n-i-i}], {i, 0, Floor[n/2]}]; A135429[n_] := A135429[n] = n!*Sum[Sum[A025035[i]*z[j]*r[n-3*i-j]/(3 i)!/j!/(n-3*i-j)!, {j, 0, n-3*i}], {i, 0, Floor[n/3]}]; A135312[n_] := SeriesCoefficient[Exp[x*Exp[x]+x^2/2], {x, 0, n}]*n!; a[n_] := A135429[n]-A135312[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz*)

Formula

a(n) = A135429(n) - A135312(n).