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A384119 Array read by antidiagonals: T(n,m) is the number of minimum dominating sets in the n X m rook graph K_n X K_m.

%I A384119 #6 May 20 2025 12:52:02
%S A384119 1,1,1,1,1,1,1,2,2,1,1,3,6,3,1,1,4,9,9,4,1,1,5,16,48,16,5,1,1,6,25,64,
%T A384119 64,25,6,1,1,7,36,125,488,125,36,7,1,1,8,49,216,625,625,216,49,8,1,1,
%U A384119 9,64,343,1296,6130,1296,343,64,9,1,1,10,81,512,2401,7776,7776,2401,512,81,10,1
%N A384119 Array read by antidiagonals: T(n,m) is the number of minimum dominating sets in the n X m rook graph K_n X K_m.
%C A384119 For m <= n, the minimum size of a dominating set is m. When m < n, solutions have exactly one vertex in each column. In the special case of n = m, solutions either have exactly one vertex in each column or have exactly one vertex in each row.
%H A384119 Andrew Howroyd, <a href="/A384119/b384119.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A384119 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>.
%H A384119 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%F A384119 T(n,m) = T(m,n).
%F A384119 T(n,m) = n^m for m < n.
%e A384119 Array begins:
%e A384119 =======================================================
%e A384119 n\m | 0 1  2   3    4     5      6       7        8 ...
%e A384119 ----+--------------------------------------------------
%e A384119   0 | 1 1  1   1    1     1      1       1        1 ...
%e A384119   1 | 1 1  2   3    4     5      6       7        8 ...
%e A384119   2 | 1 2  6   9   16    25     36      49       64 ...
%e A384119   3 | 1 3  9  48   64   125    216     343      512 ...
%e A384119   4 | 1 4 16  64  488   625   1296    2401     4096 ...
%e A384119   5 | 1 5 25 125  625  6130   7776   16807    32768 ...
%e A384119   6 | 1 6 36 216 1296  7776  92592  117649   262144 ...
%e A384119   7 | 1 7 49 343 2401 16807 117649 1642046  2097152 ...
%e A384119   8 | 1 8 64 512 4096 32768 262144 2097152 33514112 ...
%e A384119   ...
%o A384119 (PARI) T(n,m) = {if(n<=m, m^n) + if(m<=n, n^m) - if(m==n, n!)}
%Y A384119 Main diagonal is A248744.
%Y A384119 Cf. A079901, A287274, A290632.
%K A384119 nonn,tabl
%O A384119 0,8
%A A384119 _Andrew Howroyd_, May 20 2025