A290818 Array read by antidiagonals: T(m,n) = number of irredundant sets in the lattice (rook) graph K_m X K_n.

2, 3, 3, 4, 11, 4, 5, 24, 24, 5, 6, 47, 94, 47, 6, 7, 88, 272, 272, 88, 7, 8, 163, 774, 1185, 774, 163, 8, 9, 304, 2230, 4280, 4280, 2230, 304, 9, 10, 575, 6542, 15781, 20106, 15781, 6542, 575, 10, 11, 1104, 19452, 60604, 88512, 88512, 60604, 19452, 1104, 11

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Eric Weisstein's World of Mathematics, Irredundant Set

Eric Weisstein's World of Mathematics, Rook Graph

Formula

T(m,n) = A290632(m, n) + Sum_{k=0..m-1} Sum_{r=2*k..n-1} binomial(m,k) * binomial(n,r) * k! * A008299(r,k) * c(m-k,n-r) where c(m,n) = Sum_{i=0..m-1} binomial(n,i) * (n^i - n!*stirling2(i, n)).

Example

Array begins:
===============================================================
m\n| 1   2     3      4       5        6        7         8
---+-----------------------------------------------------------
1  | 2   3     4      5       6        7        8         9 ...
2  | 3  11    24     47      88      163      304       575 ...
3  | 4  24    94    272     774     2230     6542     19452 ...
4  | 5  47   272   1185    4280    15781    60604    240073 ...
5  | 6  88   774   4280   20106    88512   400728   1879744 ...
6  | 7 163  2230  15781   88512   453271  2326534  12363513 ...
7  | 8 304  6542  60604  400728  2326534 13169346  76446456 ...
8  | 9 575 19452 240073 1879744 12363513 76446456 476777153 ...
...

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}];
b[m_, n_, x_]:=m^n*x^n + n^m*x^m - If[n<=m, n!*x^m*StirlingS2[m, n], m!*x^n*StirlingS2[n, m]];
T[m_, n_]:= b[m, n, 1] + p[m, n, 1];
Table[T[n, m -n + 1], {m, 10}, {n, m}]//Flatten
(* Indranil Ghosh, Aug 12 2017, after PARI code *)

Prog

(PARI) \\ See A. Howroyd note in A290586 for explanation.
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) ))}
b(m, n, x) = m^n*x^n + n^m*x^m - if(n<=m, n!*x^m*stirling(m, n, 2), m!*x^n*stirling(n, m, 2));
T(m, n) = b(m, n, 1) + p(m, n, 1);
for(m=1, 10, for(n=1, m, print1(T(n, m-n+1), ", ")));

Crossrefs

Row 2 is A290707 for n > 1.

Main diagonal is A290586.

Cf. A287274, A290632.

Sequence in context: A128744 A293984 A207608 * A240376 A118963 A127641

Adjacent sequences: A290815 A290816 A290817 * A290819 A290820 A290821

Keyword

nonn,tabl

Author

Andrew Howroyd, Aug 11 2017

Status

approved