A290586 Number of irredundant sets in the n X n rook graph.

2, 11, 94, 1185, 20106, 453271, 13169346, 476777153, 20869990066, 1076251513071, 64077661097418, 4337014196039377, 329768528011095642, 27905789218764082151, 2608140451597365915346, 267506385903592339178241, 29943760423790270319833826

Offset: 1

Author

Eric W. Weisstein, Aug 07 2017

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100
• Andrew Howroyd, Irredundant Sets in Rook Graphs
• Eric Weisstein's World of Mathematics, Irredundant Set
• Eric Weisstein's World of Mathematics, Rook Graph

Crossrefs

Main diagonal of A290818.

Row sums of A290823.

Cf. A008299, A248744.

Programs

Mathematica

s[n_, k_]:=Sum[(-1)^i*Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}];
c[m_, n_, x_]:=Sum[Binomial[m, i] (n^i - n!*StirlingS2[i, n])*x^i, {i, 0, m - 1}];
p[m_, n_, x_]:=Sum[Sum[Binomial[m, k] Binomial[n, r]* k!*s[r, k]*x^r*c[m - k, n - r, x], {r, 2k, n - 1}], {k,0, m - 1}];
Table[2*n^n - n! + p[n, n, 1], {n, 30}]
(* Indranil Ghosh, Aug 12 2017, after PARI code *)

PARI

\\ here s(n,k) is A008299, 2*n^n - n! is A248744.
s(n,k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
c(m,n,x)=sum(i=0, m-1, binomial(m, i) * (n^i - n!*stirling(i, n, 2))*x^i);
p(m,n,x)={sum(k=0, m-1, sum(r=2*k, n-1, binomial(m,k) * binomial(n,r) * k! * s(r,k) * x^r * c(m-k,n-r,x) ))}
a(n) = 2*n^n - n! + p(n,n,1); \\ Andrew Howroyd, Aug 11 2017

Formula

a(n) = 2*n^n - n! + Sum_{k=0..n-1} Sum_{r=2*k..n-1} binomial(n,k) * binomial(n,r) * k! * A008299(r,k) * c(n-k,n-r) where c(m,n) = Sum_{i=0..m-1} binomial(n,i) * (n^i - n!*stirling2(i, n)). - Andrew Howroyd, Aug 11 2017

Extensions

a(4) corrected and a(5) from Andrew Howroyd, Aug 07 2017

Terms a(6) and beyond from Andrew Howroyd, Aug 11 2017