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A290632 Array read by antidiagonals: T(m,n) = number of minimal dominating sets in the rook graph K_m X K_n.

%I A290632 #13 Feb 16 2025 08:33:50
%S A290632 1,2,2,3,6,3,4,11,11,4,5,18,48,18,5,6,27,109,109,27,6,7,38,218,488,
%T A290632 218,38,7,8,51,405,1409,1409,405,51,8,9,66,724,3832,6130,3832,724,66,
%U A290632 9,10,83,1277,10385,21601,21601,10385,1277,83,10
%N A290632 Array read by antidiagonals: T(m,n) = number of minimal dominating sets in the rook graph K_m X K_n.
%H A290632 Andrew Howroyd, <a href="/A290632/b290632.txt">Table of n, a(n) for n = 1..1275</a>
%H A290632 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalDominatingSet.html">Minimal Dominating Set</a>
%H A290632 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%F A290632 T(m, n) = T(n, m).
%F A290632 T(n, k) = k^n + n^k - k! * stirling2(n,k) for k<=n.
%e A290632 Array begins:
%e A290632 ========================================================
%e A290632 m\n| 1  2    3     4      5       6       7        8
%e A290632 ---|----------------------------------------------------
%e A290632 1  | 1  2    3     4      5       6       7        8 ...
%e A290632 2  | 2  6   11    18     27      38      51       66 ...
%e A290632 3  | 3 11   48   109    218     405     724     1277 ...
%e A290632 4  | 4 18  109   488   1409    3832   10385    28808 ...
%e A290632 5  | 5 27  218  1409   6130   21601   78132   297393 ...
%e A290632 6  | 6 38  405  3832  21601   92592  382465  1750240 ...
%e A290632 7  | 7 51  724 10385  78132  382465 1642046  7720833 ...
%e A290632 8  | 8 66 1277 28808 297393 1750240 7720833 33514112 ...
%e A290632 ...
%t A290632 T[m_, n_] := m^n + n^m - Min[m, n]! StirlingS2[Max[m, n], Min[m, n]] (* _Eric W. Weisstein_, Aug 10 2017 *)
%o A290632 (PARI)
%o A290632 T(m,n) = m^n + n^m - if(n<=m, n!*stirling(m,n,2), m!*stirling(n,m,2));
%Y A290632 Main diagonal is A248744.
%Y A290632 Cf. A287274.
%K A290632 nonn,tabl
%O A290632 1,2
%A A290632 _Andrew Howroyd_, Aug 07 2017