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The Problem of Kings
Abstract
Let f(n) denote the number of configurations of n 2 mutually non-attacking kings on a 2n Theta 2n chessboard. We show that log f(n) grows like 2n log n Gamma 2n log 2, with an error term of O(n 4=5 log n). The result depends on an estimate for the sum of the entries of a high power of a matrix with positive entries. In chess, two kings can attack one another if their squares are horizontally, vertically, or diagonally adjacent. Consider the problem of placing mutually non-attacking kings on a chessboard with 2m rows and 2n columns. Partitioning the chessboard into 2 Theta 2 cells, we see that no cell can contain more than one king, so there can be no more than mn kings: Figure 1 In this note, we estimate the number K(m;n) of configurations of mn kings. H. Wilf [5] has obtained good estimates in the case that m is fixed and n AE m. We consider the order of growth when both m and n tend to infinity, and especially the case m = n. Our main result is stated at the end of the pape...
... This problem is similar in flavor to the "Problem of the Kings" discussed in [5] and subsequently [2], which study essentially the same problem, only with diagonal adjacencies also prohibited. Those papers focus specifically on placing k = mn nonattacking kings on an (2m) × (2n) chessboard. ...
... = T (2, 2k; k) − T (2, 2k; k + 1) by (2) and that concludes the proof of (3). ...
Using T(m,n;k) to denote the number of ways to make a selection of k squares
from an (m x n) rectangular grid with no two squares in the selection adjacent,
we give a formula for T(2,n;k), prove some identities satisfied by these
numbers, and show that T(2,n;k) is given by a degree k polynomial in n. We give
simple formulas for the first few (most significant) coefficients of the
polynomials. We give corresponding results for T(3,n;k) as well. Finally we
prove a unimodality theorem which shows, in particular, how to choose k in
order to maximize T(2,n;k).
... Likewise, the analytic capacity in [15], [19], [26] gives the growth rate of the number of configurations of mutually nonattacking princes on a hexagonal chessboard. The growth rates of the number of certain configurations of mutually nonattacking chess pieces on an chessboard have been extensively studied (e.g., for kings, in [18], [30]). The capacity calculations in [4] were formulated in terms of counting independent sets of vertices in graphs. ...
... Since different collections of of the -translates might yield the same set of points being labeled , some -valid labelings may be counted more than once in this manner. Thus, (20) where (19) follows from (18). ...
A checkerboard constraint is a bounded measurable set S⊂R2, containing the origin. A binary labeling of the Z2 lattice satisfies the checkerboard constraint S if whenever t∈Z2 is labeled 1, all of the other Z2-lattice points in the translate t+S are labeled 0. Two-dimensional channels that only allow labelings of Z2 satisfying checkerboard constraints are studied. Let A(S) be the area of S, and let A(S)→∞ mean that S retains its shape but is inflated in size in the form αS, as α→∞. It is shown that for any open checkerboard constraint S, there exist positive reals K1 and K2 such that as A(S)→∞, the channel capacity CS decays to zero at least as fast as (K1log2A(S))/A(S) and at most as fast as (K2log2A(S))/A(S). It is also shown that if S is an open convex and symmetric checkerboard constraint, then as A(S)→∞, the capacity decays exactly at the rate 4δ(S)(log2A(S))/A(S), where δ(S) is the packing density of the set S. An implication is that the capacity of such checkerboard constrained channels is asymptotically determined only by the areas of the constraint and the smallest (possibly degenerate) hexagon that can be circumscribed about the constraint. In particular, this establishes that channels with square, diamond, or hexagonal checkerboard constraints all asymptotically have the same capacity, since δ(S)=1 for such constraints.
... Slightly different two-dimensional constraints were studied for the purpose of determining the growth rates of the number of certain chess configurations (e.g. [20], [38]). In addition to run length constraints, other types of constraints have been studied as well [1], [9], [10], [11], [13], [23], [31], [32], [33], [35], [36], [37]. ...
... Finally, È ´ ¾ µ Ú ´ ¿ µ Ú È ´¼ AE ½µ Ú ´½ AE ½µ Ú (21) È ´¼ AEµ Ú ´½ AEµ Ú (22) where (21) follows from the induction hypothesis, and Condition (i) implies (22). Combining (18) with (19), (20), and (22) yields Proof. The lemma has been proved for ¾ ½ Ò Ê ¼ in Lemma 5.3. ...
A binary labeling of the triangles in a regular tiling of the two-dimensional plane satisfies the hard-triangle constraint if every triangle labeled with 1 has its three neighbors labeled with 0s. We show that the capacity associated with this constraint lies in the interval [0:628831217; 0:634775895]. The upper bound is obtained by bounding the largest magnitude eigenvalue of a certain transfer matrix and the lower bound is established by constructing an encoding algorithm whose coding rate is within 1% of the capacity.
... As we will now only be considering even-length boards, note that for the rest of the discussion n will always refer to the half-length of the even-length chessboard. The question of enumerating maximum independent arrangements of kings on the even-length 2n × 2n chessboard has been studied by Knuth [8] in an unpublished work and later by Wilf [16], and then Larsen [11] who gives an asymptotic approximation. Entry A018807 in the Online Encyclopedia of Integer Sequences [14] contributes counts up to n = 26, and Kotěšovec [9] also provides enumerative results in his extensive book on independent chessboard arrangements. ...
To count the number of maximum independent arrangements of n 2 kings on a 2n × 2n chessboard , we build a 2 n × (n + 1) matrix whose entries are independent arrangements of n kings on 2 × 2n rectangles. Utilizing upper and lower bound functions dependent of the entries of the matrix, we recursively construct independent solutions, and provide a straightforward formula and algorithm.
... In the case of queens, the number of maximum independent arrangements is unknown in general, with the most recent count for n = 27 generated through long-running computations, see Prueßer and Engelhardt [8]. For kings an asymptotic approximation is given by Larson [5], but an exact value is also unknown for even-length chessboards. Table 1 illustrates the known enumerative results including those shown in this paper, that is, here we wish to enumerate the number of maximum arrangements of nonattacking pawns. ...
Using a bijective proof, we show the number of ways to arrange a maximum number of nonattacking pawns on a chessboard is , and more generally, the number of ways to arrange a maximum number of nonattacking pawns on a chessboard is .
... If one replaces Princes by Kings on the chessboard with square cells, then the corresponding constant [34] is 1.342643951124.... A related problem, that of enumerating the maximal configurations of N 4 nonattacking Kings, was discussed in [51], [30]. ...
This is a brief survey of certain constants associated with random lattice models, including self-avoiding walks, polyominoes, the Lenz-Ising model, monomers and dimers, ice models, hard squares and hexagons, and percolation models.
A checkerboard constraint is a bounded measurable set S subset of R-2, containing the origin. A binary labeling of the Z(2) lattice satisfies the checkerboard constraint S if whenever t is an element of Z(2) is labeled 1, all of the other Z(2)-lattice points in the translate t + S are labeled 0. Two-dimensional channels that only allow labelings of Z(2) satisfying checkerboard constraints are studied. Let A(S) be the area of S, and let A(S) --> infinity mean that S retains its shape but is inflated in size in the form alphaS, as alpha --> infinity. It is shown that for any open checkerboard constraint S, there exist positive reals K-1 and K-2 such that as A(S) --> infinity, the channel capacity CS decays to zero at least as fast as (K-1 log(2) A(S))/A(S) and at most as fast as (K-2 log(2) A(S))/A(S). It is also shown that if S is an open convex and symmetric checkerboard constraint, then as A(S) --> infinity, the capacity decays exactly at the rate 4delta(S) (log(2) A (S)) /A (S), where delta(S) is the packing density of the set S. An implication. is that the capacity of such checkerboard constrained channels is asymptotically determined only by the areas of the constraint and the smallest (possibly degenerate) hexagon that can be circumscribed about the constraint. In particular, this establishes that channels with square, diamond, or hexagonal checkerboard constraints all asymptotically have the same capacity, since delta(S) = 1 for such constraints.
We continue the study of the family of planar regions dubbed Aztec diamonds in our earlier article and study the ways in which these regions can be tiled by dominoes. Two more proofs of the main formula are given. The first uses the representation theory of GL(n). The second is more combinatorial and produces a generating function that gives not only the number of domino tilings of the Aztec diamond of order n but also information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Lastly, we explore a connection between the combinatorial objects studied in this paper and the square-ice model studied by Lieb.