A378561 Number of ways to place k nonattacking anassas on an n X n chess board. Triangle T(n,k) read by rows.

1, 1, 1, 1, 4, 3, 1, 9, 22, 14, 1, 16, 82, 156, 90, 1, 25, 220, 840, 1366, 738, 1, 36, 485, 3100, 9796, 14288, 7364, 1, 49, 938, 9030, 46816, 129360, 174112, 86608, 1, 64, 1652, 22344, 172116, 767424, 1916776, 2424880, 1173240, 1, 81, 2712, 49056, 525756, 3442740, 13682320, 31572720, 38019496, 17990600

Offset: 0

Author

Eder G. Santos, Nov 30 2024

Keywords

• nonn
• easy
• tabl

Comments

Anassas (also called semi-rook+semi-bishop) are chess pieces with 2 moves: one horizontal or vertical and one diagonal.

Examples

			Triangle begins:
  1;
  1  1;
  1  4   3;
  1  9  22   14;
  1 16  82  156   90;
  1 25 220  840 1366   738;
  1 36 485 3100 9796 14288 7364;
  ...
		

Links

• 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. 716-722.
• Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024.

Crossrefs

Columns k=0-1 give: A000012, A000290.

Main diagonal gives A088789(n+1).

Programs

SageMath

print([sum([factorial(j)*binomial(n-k+j,j)*stirling_number2(n,n-k+j)*2^(k-2*j)*(binomial(k-j,j-1)+binomial(k-j+1,j)) for j in [0..ceil(k/2)]]) for n in [0..10] for k in [0..n]])

Formula

T(n,k) = Sum_{j=0..ceiling(k/2)} j! * binomial(n-k+j,j) * Stirling2(n,n-k+j) * 2^(k-2*j) * (binomial(k-j,j-1) + binomial(k-j+1,j)).