This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384123 #10 May 22 2025 16:57:28
%S A384123 1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,5,5,1,1,1,1,12,48,12,1,1,1,1,37,
%T A384123 121,121,37,1,1,1,1,98,278,320,278,98,1,1,1,1,219,579,729,729,579,219,
%U A384123 1,1,1,1,430,1102,1480,1610,1480,1102,430,1,1,1,1,767,1943,2741,3161,3161,2741,1943,767,1,1
%N A384123 Array read by antidiagonals: T(n,m) is the number of minimal dominating sets in the n X m rook complement graph.
%C A384123 For n > 2, m > 2, the minimal dominating sets are:
%C A384123 - all vertices in any single row or column,
%C A384123 - any three vertices such that no two are in the same row or column,
%C A384123 - any vertex with another in the same row and a third in the same column,
%C A384123 - two vertices in each of two rows/columns and none in the same column/row.
%C A384123 Except for (n,m) = (2,3) or (3,2), also the number of maximal irredundant sets in the n X m rook complement graph. In particular, there are 11 maximal irredundant sets in these two graphs.
%H A384123 Andrew Howroyd, <a href="/A384123/b384123.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A384123 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>.
%H A384123 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalDominatingSet.html">Minimal Dominating Set</a>.
%H A384123 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>.
%F A384123 T(n,m) = n + m + 6*binomial(n,3)*binomial(m,3) + n*(n-1)*m*(m-1) + 6*binomial(n,4)*binomial(m,2) + 6*binomial(n,2)*binomial(m,4) for n > 2, m > 2.
%F A384123 T(n,m) = T(m,n).
%e A384123 Array begins:
%e A384123 ===================================================
%e A384123 n\m | 0 1 2 3 4 5 6 7 8 ...
%e A384123 ----+---------------------------------------------
%e A384123 0 | 1 1 1 1 1 1 1 1 1 ...
%e A384123 1 | 1 1 1 1 1 1 1 1 1 ...
%e A384123 2 | 1 1 4 5 12 37 98 219 430 ...
%e A384123 3 | 1 1 5 48 121 278 579 1102 1943 ...
%e A384123 4 | 1 1 12 121 320 729 1480 2741 4716 ...
%e A384123 5 | 1 1 37 278 729 1610 3161 5682 9533 ...
%e A384123 6 | 1 1 98 579 1480 3161 6012 10513 17234 ...
%e A384123 7 | 1 1 219 1102 2741 5682 10513 17948 28827 ...
%e A384123 8 | 1 1 430 1943 4716 9533 17234 28827 45488 ...
%e A384123 ...
%e A384123 The T(2,3) = 5 minimal dominating sets are those that contain all vertices in either a single row or a single column. There are also 6 maximal irredundant sets that are not dominating. These are those that contain one vertex in each of the two rows but not in the same column.
%o A384123 (PARI) T(n,m) = {if(n<=1||m<=1, 1, n + m + 6*binomial(n,3)*binomial(m,3) + if(n > 2 && m > 2, n*(n-1)*m*(m-1)) + 6*binomial(n,4)*binomial(m,2) + 6*binomial(n,2)*binomial(m,4))}
%Y A384123 Main diagonal is A291623.
%Y A384123 Columns 0 and 1 are A000012.
%Y A384123 Column 2 is A289121 for n > 1.
%Y A384123 Cf. A384121, A384122.
%K A384123 nonn,tabl
%O A384123 0,13
%A A384123 _Andrew Howroyd_, May 20 2025