Counting Nonattacking Chess Piece Placements: Bishops and Anassas

We derive recurrences and closed-form expressions for counting nonattacking placements of two types of chess pieces with unbounded straight-line moves, namely the bishop (two diagonal moves) and the anassa (one horizontal or vertical move and one diagonal move), placed on a standard square chessboard. Additionally, we obtain explicit expressions for the corresponding quasi-polynomial coefficients. The recurrences are derived by analyzing how nonattacking configurations attack a specific subset of board squares, employing a bijective argument to establish the relations. The main results are simplifications of known expressions for the bishop and a general counting formula for the anassa.

💡 Research Summary

The paper investigates the enumeration of non‑attacking placements of two types of “rider” chess pieces—bishops (two diagonal moves) and anassas (one orthogonal and one diagonal move)—on an m × m square board. The authors introduce a general combinatorial framework based on “collapsibility” and “inductive subsets”. A board Bₘ together with a chosen subset Iₘ is said to be collapsible under a move set M_P if there exists a bijection f from Bₘ \ Iₘ onto Bₘ₋₁ that preserves the adjacency relations induced by M_P. Theorem 1.2 shows that the number of non‑attacking configurations on Bₘ \ Iₘ equals the number on the smaller board Bₘ₋₁, providing a systematic way to reduce the problem size.

For bishops, the move set is M_B = {(1,1), (−1,1)}. By separating the board into white and black squares, the problem splits into two independent rook‑placement problems on transformed boards Wₘ and Kₘ. An explicit inductive subset Iₘ (the shaded columns in Figure 1) yields a bijection f_B that shifts the upper part of the board down one row and the lower part left one column, proving collapsibility. This leads to the recurrences

R_W(m,k)=R_W(m−1,k)+(m−k+π(m))R_W(m−1,k−1)

R_K(m,k)=R_K(m−1,k)+(m−k+1−π(m))R_K(m−1,k−1)

where π(m)=1 for odd m and 0 otherwise, together with the natural initial conditions. Solving these recurrences gives closed‑form expressions involving binomial coefficients:

R_W(m,k)=∑_{j=0}^{⌊m/2⌋} C(m−j, m−k)·C(⌊m/2⌋, j)

R_K(m,k)=∑_{j=0}^{⌊m/2⌋} C(m−j, m−k)·C(⌈m/2⌉, j).

The total number of bishop placements is then

B_S(m,k)=∑_{j=0}^{k} R_W(m,k−j)·R_K(m,j)

which can be rewritten as a triple sum (Equation 5). This expression is manifestly a quasi‑polynomial of degree 2k in m; the period is shown to be t = 2 for all k ≥ 3, while for k = 1, 2 the period collapses to 1. Substituting m = −1 yields B_S(−1,k)=k!, confirming a known result that the number of combinatorial types of non‑attacking configurations equals k! for any rider with exactly two unbounded directions.

To obtain explicit coefficients of the quasi‑polynomials, the authors prove Lemma 2.6 (a binomial‑Stirling identity) and Lemma 2.7 (a decomposition of products of shifted binomials into a sum involving β_i(p,q,z)). Using these lemmas they derive formulas (13) and (14) for the coefficients