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A291543
Array read by antidiagonals: T(m,n) = number of maximal irredundant sets in the lattice (rook) graph K_m X K_n.
2
1, 2, 2, 3, 6, 3, 4, 11, 11, 4, 5, 18, 48, 18, 5, 6, 27, 109, 109, 27, 6, 7, 38, 218, 632, 218, 38, 7, 8, 51, 405, 1649, 1649, 405, 51, 8, 9, 66, 724, 4192, 10130, 4192, 724, 66, 9, 10, 83, 1277, 10889, 34801, 34801, 10889, 1277, 83, 10
OFFSET
1,2
COMMENTS
Maximal irredundant sets can be either dominating or not. The dominating maximal irredundant sets are the minimal dominating sets (A290632). The non-dominating maximal irredundant sets are those irredundant sets that have exactly one empty row and one empty column and at least one row and one column with more than one element. See note in A290586 for some additional details.
LINKS
Eric Weisstein's World of Mathematics, Maximal Irredundant Set
Eric Weisstein's World of Mathematics, Rook Graph
FORMULA
T(m,n) = A290632(m, n) + Sum_{k=2..m-2} Sum_{j=2..m-k} binomial(m,k) * binomial(n,j) * k! * A008299(n-j,k-1) * j! * stirling2(m-k,j-1).
EXAMPLE
Array begins:
=========================================================
m\n| 1 2 3 4 5 6 7 8
---|-----------------------------------------------------
1 | 1 2 3 4 5 6 7 8...
2 | 2 6 11 18 27 38 51 66...
3 | 3 11 48 109 218 405 724 1277...
4 | 4 18 109 632 1649 4192 10889 29480...
5 | 5 27 218 1649 10130 34801 116772 402673...
6 | 6 38 405 4192 34801 194292 856225 3804880...
7 | 7 51 724 10889 116772 856225 4730810 24810465...
8 | 8 66 1277 29480 402673 3804880 24810465 145114944...
...
MATHEMATICA
T32[n_, k_] := n^k + k^n - Min[n, k]!*StirlingS2[Max[n, k], Min[n, k]];
T99[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*((k - i - j)^(n - i)/(j!*(k - i - j)!)), {j, 0, k - i}], {i, 0, k}];
T[m_, n_] /; n >= m := T32[m, n] + Sum[Sum[Binomial[m, k]*Binomial[n, j]*k!*T99[n - j, k - 1]*j!*StirlingS2[m - k, j - 1], {j, 2, m - k}], {k, 2, m - 2}]; T[m_, n_] /; n < m := T[n, m];
Table[T[m - n + 1, n], {m, 1, 10}, {n, 1, m}] // Flatten(* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
PROG
(PARI) \\ here s(n, k) is A008299(n, k) and b(m, n, 1) is A290632(m, n).
s(n, k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
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));
p(m, n, x)={sum(k=2, m-2, sum(j=2, m-k, binomial(m, k) * binomial(n, j) * k! * s(n-j, k-1) * j! * stirling(m-k, j-1, 2) * x^(m+n-j-k) ))}
T(m, n) = b(m, n, 1) + p(m, n, 1);
CROSSREFS
Main diagonal is A291104.
Sequence in context: A272973 A272958 A290632 * A296320 A296396 A125102
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Aug 25 2017
STATUS
approved