A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).

1, 1, 1, 2, 1, 4, 2, 1, 8, 14, 4, 1, 12, 38, 32, 4, 1, 18, 98, 184, 100, 8, 1, 24, 188, 576, 652, 208, 8, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 40, 580, 3840, 12052, 16944, 9080, 1280, 16, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32

Offset: 0

Author

N. J. A. Sloane, Jun 14 2016

Keywords

• nonn
• tabf

Comments

From Eder G. Santos, Dec 16 2024: (Start)

The sequence counts every possible nonattacking configuration of k bishops on the white squares of an n X n chess board.

It is assumed that the n X n chess board has a black square in the upper left corner.

(End)

Examples

			Triangle begins:
  1;
  1;
  1,  2;
  1,  4,    2;
  1,  8,   14,     4;
  1, 12,   38,    32,     4;
  1, 18,   98,   184,   100,      8;
  1, 24,  188,   576,   652,    208,      8;
  1, 32,  356,  1704,  3532,   2816,    632,     16;
  1, 40,  580,  3840, 12052,  16944,   9080,   1280,     16;
  1, 50,  940,  8480, 38932,  89256,  93800,  37600,   3856,   32;
  1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32;
  ...
From _Eder G. Santos_, Dec 16 2024: (Start)
For example, for n = 3, k = 2, the T(3,2) = 2 nonattacking configurations are:
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
  |   |   |   | , | B |   | B |
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
(End)
		

Links

• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. See Section 9.
• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. See Section 9. [Annotated scanned copy]
• J. Perott, Sur le problème des fous, Bulletin de la S. M. F., tome 11 (1883), pp. 173-186.
• Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas. arXiv:2411.16492 [math.CO], 2024. (considered as black board).
• Eric Weisstein's World of Mathematics, White Bishop Graph.

Crossrefs

Columns k=0-1 give: A000012, A007590.

Alternate rows give A088960.

Row sums are A216078(n+1).

T(2n,n) gives A191236.

T(2n+1,n) gives A217900(n+1).

T(n+1,n) gives A060546.

Cf. A274105 (black squares), A288182, A201862, A002465.

Programs

Maple

with(combinat): with(gfun):
T := n -> add(stirling2(n+1,n+1-k)*x^k, k=0..n):
# bishops on white squares
bish := proc(n) local m,k,i,j,t1,t2; global T;
    if n=0 then return [1] fi;
    if (n mod 2) = 0 then m:=n/2;
        t1:=add(binomial(m,k)*T(2*m-1-k)*x^k, k=0..m);
    else
        m:=(n-1)/2;
        t1:=add(binomial(m,k)*T(2*m-k)*x^k, k=0..m+1);
    fi;
    seriestolist(series(t1,x,2*n+1));
end:
for n from 0 to 12 do lprint(bish(n)); od:

Mathematica

T[n_] := Sum[StirlingS2[n+1, n+1-k]*x^k, {k, 0, n}];
bish[n_] := Module[{m, t1, t2}, If[Mod[n, 2] == 0,
   m = n/2;     t1 = Sum[Binomial[m, k]*T[2*m-1-k]*x^k, {k, 0, m}],
   m = (n-1)/2; t1 = Sum[Binomial[m, k]*T[2*m - k]*x^k, {k, 0, m+1}]];
CoefficientList[t1 + O[x]^(2*n+1), x]];
Table[bish[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 25 2022, after Maple code *)

SageMath

def stirling2_negativek(n, k):
  if k < 0: return 0
  else: return stirling_number2(n, k)
print([sum([binomial(floor(n/2), j)*stirling2_negativek(n-j, n-k) for j in [0..k]]) for n in [0..10] for k in [0..n-1+kronecker_delta(n,0)]]) # Eder G. Santos, Dec 01 2024

Formula

From Eder G. Santos, Dec 01 2024: (Start)

T(n,k) = Sum_{j=0..k} binomial(floor(n/2),j) * Stirling2(n-j,n-k).

T(n,k) = T(n-1,k) + (n-k+1-A000035(n)) * T(n-1,k-1), T(n,0) = 1, T(0,k) = delta(k,0). (End)

Extensions

T(0,0) prepended by Eder G. Santos, Dec 01 2024