A187242 -id:A187242 - cp's OEIS frontend

A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

1, 2, 9, 70, 729, 9918, 167281, 3423362, 82609921, 2319730026, 74500064809, 2711723081550, 110568316431609, 5016846683306758, 251180326892449969, 13806795579059621930, 827911558468860287041, 53940895144894708523922, 3799498445458163685753481, 288400498147873552894868886

Offset: 0

Vaclav Kotesovec, Dec 06 2011

nonn

Also the number of vertex covers and independent vertex sets of the n X n bishop graph.

Vaclav Kotesovec, Table of n, a(n) for n = 0..320
Vaclav Kotesovec, Non-attacking chess pieces
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A010790, A172123, A172124, A172127, A172129, A176886, A187239 - A187242, A002465, A216078, A216332.

Row sums of A378590.

knbishops[k_,n_]:=(If[n==1,If[k==1,1,0],(-1)^k/(2n-k)!
*Sum[Binomial[2n-k,n-k+i]*Sum[(-1)^m*Binomial[n-i,m]*m^Floor[n/2]*(m+1)^Floor[(n+1)/2],{m,1,n-i}]
*Sum[(-1)^m*Binomial[n-k+i,m]*m^Floor[(n+1)/2]*(m+1)^Floor[n/2],{m,1,n+i-k}],{i,Max[0,k-n],Min[k,n]}]]);
Table[1+Sum[knbishops[k,n],{k,1,2n-1}],{n,1,25}]

a(n) = A216078(n+1) * A216332(n+1). - Andrew Howroyd, May 08 2017

a(0)=1 prepended by Alois P. Heinz, Dec 01 2024

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]).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 26, 26, 8, 1, 16, 92, 232, 260, 112, 16, 1, 25, 240, 1124, 2728, 3368, 1960, 440, 32, 1, 36, 520, 3896, 16428, 39680, 53744, 38368, 12944, 1600, 64, 1, 49, 994, 10894, 70792, 282248, 692320, 1022320, 867328, 389312, 81184, 5792, 128

Offset: 0

Eder G. Santos, Dec 01 2024

The sequence counts every possible nonattacking configuration of k bishops on an n x n chess board.

			Triangle begins:
  1;
  1  1;
  1  4   4;
  1  9  26    26     8;
  1 16  92   232   260    112     16;
  1 25 240  1124  2728   3368   1960     440     32;
  1 36 520  3896 16428  39680  53744   38368  12944   1600    64;
  1 49 994 10894 70792 282248 692320 1022320 867328 389312 81184 5792 128;
  ...
For example, for n = 2, k=2, the T(2,2)=4 nonattacking configurations are:
  +---+---+   +---+---+   +---+---+   +---+---+
  | B | B |   | B |   |   |   | B |   |   |   |
  +---+---+ , +---+---+ , +---+---+ , +---+---+
  |   |   |   | B |   |   |   | B |   | B | B |
  +---+---+   +---+---+   +---+---+   +---+---+

S. Chaiken, C. R. H. Hanusa, and T. Zaslavsky, A q-Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders, J. Korean Math. Soc., 57(6): 1407-1433, 2020; see also arXiv preprint, arXiv:1609.00853 [math.CO], 2016-2020.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 234-259.
Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024.

Columns k=0-1 give: A000012, A000290.
Columns k=2-10 for n>=1 give: A172123, A172124, A172127, A172129, A176886, A187239, A187240, A187241, A187242.
Main diagonal T(n,n) gives A002465.
Row sums give A201862.
Cf. A000079.

def stirling2_negativek(n,k):
  if k < 0: return 0
  else: return stirling_number2(n,k)
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)]])

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)).

T(n,2*n-2+delta(n,1)+2*delta(n,0)) = A000079(n)-delta(n,1).

cp's OEIS Frontend

A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions