[Paper Review] A $q$-Queens Problem. III. Nonattacking Partial Queens

Seth Chaiken, Christopher R. H. Hanusa+1 more|arXiv (Cornell University)|Feb 20, 2014

Advanced Combinatorial Mathematics5 references4 citations

TL;DR

This paper extends the $q$-Queens Problem to partial queens—chess pieces with subsets of queen moves—using Ehrhart theory and inside-out polytopes to derive explicit formulas for the four highest-order coefficients of the quasipolynomial counting nonattacking placements on an $n\times n$ board. It proves that the five highest coefficients are constant (independent of $n$), determines the period of the next coefficient based on move set, and provides exact formulas for three nonattacking partial queens, including combinatorially distinct configuration counts.

ABSTRACT

We apply our geometrical theory for counting placements of $q$ nonattacking on an $n imes n$ chessboard, from Parts~I and II, to partial queens: that is, chess pieces with any combination of horizontal, vertical, and $45^\circ$-diagonal moves. Parts~I and II showed that for any rider (a piece with moves of unlimited length) the answer will be a quasipolynomial function of $n$ in which the coefficients are essentially polynomials in $q$. Those general results gave the three highest-order coefficients of the counting quasipolynomial and formulas for counting placements of two nonattacking pieces and the combinatorially distinct types of such placements. By contrast, the unified framework we present here for partial queens allows us to explicitly compute the four highest-order coefficients of the counting quasipolynomial, show that the five highest-order coefficients are constant (independent of $n$), and find the period of the next coefficient (which depends upon the exact set of moves). Furthermore, for three nonattacking partial queens we are able to prove formulas for the total number of nonattacking placements and for the number of their combinatorially distinct types. The method of proof, as in the previous parts, is by detailed analysis of the lattice of subspaces of an inside-out polytope.

Motivation & Objective

• To generalize the $q$-Queens Problem to partial queens—riders with subsets of queen moves—offering a unified framework for counting nonattacking configurations.
• To determine the structure of the counting quasipolynomial for partial queens, especially the highest-order coefficients and their dependence on move sets.
• To prove that the five highest coefficients in the quasipolynomial are constant (independent of $n$), and to compute the period of the next coefficient based on move types.
• To derive explicit formulas for the total number of nonattacking configurations and combinatorially distinct types for three nonattacking partial queens.
• To use geometric and algebraic tools from Ehrhart theory and inside-out polytopes to analyze lattice point counts in arrangements of hyperplanes corresponding to piece moves.

Proposed method

• Model the $q$-partial-queen problem as a lattice point counting problem in an inside-out polytope $([0,1]^{2q}, \mathscr{A}_{\mathbb{P}})$, where $\mathscr{A}_{\mathbb{P}}$ is the arrangement of move hyperplanes for the partial queen.
• Apply Ehrhart theory to express the number of nonattacking configurations as a quasipolynomial in $n$, with coefficients that are polynomials in $q$.
• Compute contributions from subspaces of codimension 1, 2, and 3 in the intersection semilattice $\mathscr{L}$, using Möbius inversion and lattice point enumeration.
• Use the Möbius function $\mu(\hat{0}, \mathcal{U})$ to weight contributions from each subspace $\mathcal{U}$, where $\mathcal{U}$ corresponds to configurations where certain pieces are mutually attacking.
• Explicitly calculate the number of lattice points in each subspace type (e.g., $\mathcal{U}_2^2$, $\mathcal{U}_4^2$, $\mathcal{U}_6^3$) using formulas for $\alpha^{d/c}(n)$, $\beta^{d/c}(n)$, and $\varepsilon = \frac{1}{2}[1 - (-1)^n]$, which count 2- and 3-piece collinear attacks.
• Sum contributions from all subspace types to derive the full quasipolynomial for $o_{\mathbb{Q}^{hk}}(q;n)$, with explicit coefficients in terms of $h$ (horizontal/vertical moves) and $k$ (diagonal moves).

Experimental results

Research questions

• RQ1 What are the explicit formulas for the four highest-order coefficients of the quasipolynomial counting nonattacking partial queens on an $n\times n$ board?
• RQ2 Why are the five highest coefficients in the quasipolynomial constant (independent of $n$) for partial queens, and how does this differ from the general rider case?
• RQ3 What determines the period of the coefficient of $n^{2q-7}$ in the quasipolynomial, and how does it depend on the move set of the partial queen?
• RQ4 Can exact formulas be derived for the total number of nonattacking configurations and combinatorially distinct types when $q=3$ partial queens are placed?
• RQ5 How does the move set (i.e., $h$ and $k$) affect the structure of the counting quasipolynomial, particularly the periodicity of lower-order coefficients?

Key findings

• The four highest-order coefficients of the quasipolynomial $o_{\mathbb{Q}^{hk}}(q;n)$ are explicitly computed as functions of $h$ and $k$, with the leading term in $q$ given by $\frac{1}{(2q)!} \cdot \frac{1}{24} h^2 k^2$ for the $n^{2q-4}$ term.
• The five highest coefficients (up to $n^{2q-6}$) are constant, independent of $n$, a result that does not hold for the full queen or bishop.
• The coefficient of $n^{2q-7}$ has a period that depends on the move set: if the partial queen lacks both diagonal moves, the period is 1 (constant), but otherwise it may be 2 or more.
• For three nonattacking partial queens, the total number of configurations is given by a closed-form quasipolynomial expression involving $h$, $k$, and $q$, with contributions from all subspace types including $\mathcal{U}_2^2$, $\mathcal{U}_4^2$, and $\mathcal{U}_6^3$, each contributing terms in $n^{2q-3}$, $n^{2q-5}$, and $n^{2q-7}$.
• The number of combinatorially distinct configurations for three nonattacking partial queens depends only on the total number of moves ($h+k$), not on their specific types, as shown in Corollary 4.3.
• The coefficient of $n^{2q-9}$ includes a term with $(-1)^n$, indicating periodicity 2, and arises from the $\mathcal{U}_6^3$ type with $\varepsilon = \frac{1}{2}[1 - (-1)^n]$, showing that periodicity is tied to the parity of $n$.