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A378590 Total number of ways to place k nonattacking bishops on an n X n chess board. Triangle T(n,k) read by rows (0 <= k <= 2*n-[n>0]-[n>1]).

%I A378590 #24 Dec 28 2024 10:19:37
%S A378590 1,1,1,1,4,4,1,9,26,26,8,1,16,92,232,260,112,16,1,25,240,1124,2728,
%T A378590 3368,1960,440,32,1,36,520,3896,16428,39680,53744,38368,12944,1600,64,
%U A378590 1,49,994,10894,70792,282248,692320,1022320,867328,389312,81184,5792,128
%N A378590 Total number of ways to place k nonattacking bishops on an n X n chess board. Triangle T(n,k) read by rows (0 <= k <= 2*n-[n>0]-[n>1]).
%C A378590 The sequence counts every possible nonattacking configuration of k bishops on an n x n chess board.
%H A378590 S. Chaiken, C. R. H. Hanusa, and T. Zaslavsky, <a href="https://doi.org/10.4134/JKMS.j190682">A q-Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders</a>, J. Korean Math. Soc., 57(6): 1407-1433, 2020; see also <a href="https://arxiv.org/abs/1609.00853">arXiv preprint</a>, arXiv:1609.00853 [math.CO], 2016-2020.
%H A378590 Vaclav Kotesovec, <a href="http://www.kotesovec.cz/books/kotesovec_non_attacking_chess_pieces_2013_6ed.pdf">Non-attacking chess pieces</a>, 6ed, 2013, p. 234-259.
%H A378590 Eder G. Santos, <a href="https://arxiv.org/abs/2411.16492">Counting non-attacking chess pieces placements: Bishops and Anassas</a>, arXiv:2411.16492 [math.CO], 2024.
%F A378590 T(n,k) = Sum_{j=0..k} (Sum_{i=0..j} binomial(floor(n/2),i) * Stirling2(n-i,n-j)) * (Sum_{l=0..k-j} binomial(ceiling(n/2),l) * Stirling2(n-l,n-k+j)).
%F A378590 T(n,2*n-2+delta(n,1)+2*delta(n,0)) = A000079(n)-delta(n,1).
%e A378590 Triangle begins:
%e A378590   1;
%e A378590   1  1;
%e A378590   1  4   4;
%e A378590   1  9  26    26     8;
%e A378590   1 16  92   232   260    112     16;
%e A378590   1 25 240  1124  2728   3368   1960     440     32;
%e A378590   1 36 520  3896 16428  39680  53744   38368  12944   1600    64;
%e A378590   1 49 994 10894 70792 282248 692320 1022320 867328 389312 81184 5792 128;
%e A378590   ...
%e A378590 For example, for n = 2, k=2, the T(2,2)=4 nonattacking configurations are:
%e A378590   +---+---+   +---+---+   +---+---+   +---+---+
%e A378590   | B | B |   | B |   |   |   | B |   |   |   |
%e A378590   +---+---+ , +---+---+ , +---+---+ , +---+---+
%e A378590   |   |   |   | B |   |   |   | B |   | B | B |
%e A378590   +---+---+   +---+---+   +---+---+   +---+---+
%o A378590 (SageMath) def stirling2_negativek(n,k):
%o A378590   if k < 0: return 0
%o A378590   else: return stirling_number2(n,k)
%o A378590 print([sum([sum([binomial(floor(n/2),i)*stirling2_negativek(n-i,n-j)*sum([binomial(ceil(n/2),l)*stirling2_negativek(n-l,n-k+j) for l in [0..k-j]]) for i in [0..j]]) for j in [0..k]]) for n in [0..10] for k in [0..2*n-2+kronecker_delta(n,1)+2*kronecker_delta(n,0)]])
%Y A378590 Columns k=0-1 give: A000012, A000290.
%Y A378590 Columns k=2-10 for n>=1 give: A172123, A172124, A172127, A172129, A176886, A187239, A187240, A187241, A187242.
%Y A378590 Main diagonal T(n,n) gives A002465.
%Y A378590 Row sums give A201862.
%Y A378590 Cf. A000079.
%K A378590 nonn,easy,tabf
%O A378590 0,5
%A A378590 _Eder G. Santos_, Dec 01 2024