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A290724 Triangle read by rows: T(n,k) = number of arrangements of k non-attacking rooks on an n X n right triangular board with every square controlled by at least one rook.

%I A290724 #13 Feb 16 2025 08:33:50
%S A290724 1,1,1,0,4,1,0,2,11,1,0,0,18,26,1,0,0,6,100,57,1,0,0,0,96,444,120,1,0,
%T A290724 0,0,24,900,1734,247,1,0,0,0,0,600,6480,6246,502,1,0,0,0,0,120,8520,
%U A290724 39762,21320,1013,1,0,0,0,0,0,4320,90600,219312,70128,2036,1
%N A290724 Triangle read by rows: T(n,k) = number of arrangements of k non-attacking rooks on an n X n right triangular board with every square controlled by at least one rook.
%C A290724 See A146304 for algorithm and PARI code to produce this sequence.
%C A290724 Equivalently, the number of maximal independent vertex sets in the n-triangular honeycomb bishop graph with k vertices. A bishop can move along two axes in the triangular honeycomb grid.
%H A290724 Andrew Howroyd, <a href="/A290724/b290724.txt">Table of n, a(n) for n = 1..1275</a>
%H A290724 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>
%e A290724 Triangle begins:
%e A290724 1;
%e A290724 1, 1;
%e A290724 0, 4,  1;
%e A290724 0, 2, 11,  1;
%e A290724 0, 0, 18,  26,  1;
%e A290724 0, 0,  6, 100,  57,    1;
%e A290724 0, 0,  0,  96, 444,  120,     1;
%e A290724 0, 0,  0,  24, 900, 1734,   247,     1;
%e A290724 0, 0,  0,  0,  600, 6480,  6246,   502,    1;
%e A290724 0, 0,  0,  0,  120, 8520, 39762, 21320, 1013, 1;
%e A290724 ...
%t A290724 CoefficientList[Table[Sum[k! StirlingS2[m, k] StirlingS2[n + 1 - m, k + 1] x^(n - k), {m, 0, n}, {k, 0, Min[m, n - m]}], {n, 20}]/x, x] // Flatten (* _Eric W. Weisstein_, Feb 01 2024 *)
%Y A290724 Row sums are A290615.
%Y A290724 Cf. A259691, A259697.
%K A290724 nonn,tabl
%O A290724 1,5
%A A290724 _Andrew Howroyd_, Aug 09 2017