Generalisation of bishop polynomial derived from conventional rook polynomial Sim Ming Hui Melodies, Qu Siyang k non-attacking rooks on an m × n chess- Raffles Girls’ School (Secondary) board. A rook is a chess piece that moves Singapore, Singapore vertically and horizontally on the chess- E-mail: [email protected], board. As such, non-attacking rooks must [email protected] be arranged such that no rook are placed in the same row or column. 1. Abstract Let B denote a generalised board (which The rook polynomial is a powerful tool in may be irregular in shape), whereas Bm×n the study of restricted permutations. The denote a rectangular (regular) board with bishop polynomial is a subset of the rook m rows and n columns. In this paper, polynomial on irregular boards. While we define the term k-rook placement as the rook polynomial for regular boards the number of arrangements of k non- is well-established, a generalisation for attacking rooks on a board B whereby or- the bishop polynomial remains unclear der does not matter. (at the point of the study). This study The following are theorems and defini- hence presents an original generalisation, tions previously derived by other papers. the Qu-Sim theorem, for the bishop poly- Proof will not be provided here but can be nomial for all square boards. easily found in the references. 2. Introduction Definition 3.1 The rook polynomial of a board B is the generating function: The rook polynomial is a powerful tool 1 R (x) = P r (B)xk, where r (B) in the theory of restricted permutations. B k=0 k k denotes the k-rook placement on B. [1] Especially in recent years, the study of rook polynomial branches from the ini- mn Lemma 3.1 rk(Bm×n) = k k k! tial focus on the enumeration of restricted where k ≥ m; n. permutations to many others, with ap- plications in graph theory, enumeration Theorem 3.2 The rook polynomial of of matrices, chromatic theory and more. Bm×n is defined as: Rm;n(x) = In comparison, the bishop polynomial, a Pmin(m;n) mn k k=0 k k k! x while a 0-rook special case of the rook polynomial, has placement on any board is always 1. not been as well- established. As such, this paper hopes to present an original Definition 3.2 Two boards, A and B, are generalisation of the bishop polynomial said to be rook equivalent if RA(x) = for square boards to help raise the thresh- RB(x). A sufficient (while not always old of research on bishop polynomials. necessary) condition of rook equivalence is that B can be obtained from A by per- 3. Preliminaries mutation of rows and columns, meaning interchanging rows and columns will not (Note that a glossary of notations is avail- alter the k-rook placements of a board.[1] able at the Appendix for easy of refer- ence.) Theorem 3.3 Component Theorem Let B The rook polynomial is a generating poly- be a board consisting of k disjoint sub- nomial of the number of arrangements of boards, B1;B2;B3:::Bk. Then RB(x) = k Πi=1RBk (x), where disjoint sub-boards Corollary 4.0.1 The rook polynomial for are defined as boards made up of cells n × 2 boards is as follows: Rn×2(x) = 2 from the original board B not sharing 1+(n×2)x+(n)(n−1)x where Rn×2(x) common rows/columns.[2] denotes the rook polynomial for n × 2 boards. Definition 3.3 Let Bm×n be an m×n board, and S an s × t block of Bm×n. For Proof 4.0.1 : Since there is always 1 way 0 ≤ j ≤ min(s; t), let BS;j denote the to arrange 0 rooks on any board. Con- board obtained by deleting sider all possible n × 2 boards, using Theorem 2.2 there is n2 = 2n ways • j of the s rows covered by S 1 1 • j of the t columns covered by S to arrange 1 non-attacking rook; and n2 • all the cells of S 2 2 2! = (n)(n − 1) ways to arrange 2 non-attacking rooks. th BS;j is known as the j inclusion board of Bm×n relative to S. Since rows and 5. Introduction to the Bishop columns involved are identical except on Polynomial S, which are deleted, BS;j is well-defined. [1] A bishop moves only diagonally without Theorem 3.4 Block Decomposition The- restriction in the distance of each move. orem [1] Definition 5.1 Here, the bishop poly- min (s;t) nomial is the generating function of X j R(B) = rk(S)x R(BS;j) the number of arrangements of k non- j=0 attacking bishops (k-bishop placement) on a m × n board: Bm×n(x) = (Efficiency of the theorem is greatly af- P k th k=0 bkx where bk denotes the k co- fected by the choice of cell/block to de- efficient of the bishop polynomial. compose.) Example: In Figure 1, let S be the 2 × 1 While both bishops and rooks move in shaded block. Then by Theorem 2.4, straight lines, they differ in the direction the rook polynomial of un-decomposed of movement. However, the movement of the two can be related through a 45o ro- board B is r0(S)R(Bs;0) + r1(S)R(Bs;1) as shown. tation of the board. Tracing out the path of a bishop after a 45o rotation, gives the path of a rook piece. In Figure 2, the bishop polynomial of board A is the rook polynomial of board B. Example of Application of Block Decomposition 4. Special case of Rook Polynomial: 2 × n boards This special case of the Rook Polynomial will be used intensively in our proof of our main theorems presented, accompa- nied with Theorem 2.4. As such, it is instructive to go through the general for- mula of the rook polynomial for n × 2 Tilting board 45o converts bishop move (in A) to a rook boards. one (in B) 2 Given that a rook moves only vertically or 6.2. Qu-Sim Theorem for Even Boards horizontally, it can only occupy squares Theorem 6.1 B (x) = (P2n−3 of the same colour (in board B above). 2n×2n k=0 xk b (E (x))×R (x))2 Thus the white board (consisting of white k (2n−2)×(2n−2) 2×(2n−k−1) cells) and black board (consisting of black cells) are disjoint sub-boards of the over- all board B. In particular, for even boards where n ≥ 2 and E2×2(x) = 1 + 2x: Note th B2n×2n, note that R(Bwhite) = R(Bblack). that bk(B(x)) refers to the k bishop By Theorem 2.3, R(B) = R(Bwhite) × coefficient of bishop polynomial of sub- R(Bblack) for all square board B. board E(x) and Rm×n(x) refers to the rook polynomial of regular board Bm×n. 6. The Qu-Sim Theorem Proof 6.1.1 Transforming the first few even boards (Figure 4) gives us a gen- 6.1. Definitions eral idea of the pattern: The sub-boards Definition 6.1 Square even boards, E(2n+2)×(2n+2) always adds a 2×(2n+1) B2n×2n can be decomposed into two rook block to the former E sub-board E2n×2n. equivalent sub-boards, each denoted as Hence, as the consecutive boards are E board or E2n×2n. Bishop polynomial closely related, a recursive function can of E2n×2n is expressed as E2n×2n(x). be used to calculate the bishop polyno- Based on Theorem 2.3, B2n×2n(x) = mial for each board. 2 (E2n×2n(x)) Definition 6.2 Square odd boards, B(2n+1)×(2n+1) can be broken down into two sub-boards: one with even num- First few E boards (subboard of the even board ber of cells – denoted as P board or B2n×2n) P(2n+1)×(2n+1) - and the other with an odd Using Theorem 2.4, we see that for number of cells – denoted as O board or E2n×2n the following relations hold if O(2n+1)×(2n+1). we choose to decompose the previous E(2n−2)×(2n−2) board: (Refer to Figure 5 Bishop polynomial of the P and O for diagrammatic representation of rela- boards are expressed as P(2n+1)×(2n+1)(x) tions.) and O(2n+1)×(2n+1)(x). By Theorem 2.3, B(2n+1)×(2n+1)(x) = P(2n+1)×(2n+1)(x) × O(2n+1)×(2n+1)(x) Block Decomposition of the E2n×2n board) (which is the board on the LHS of equation) Representing Figure 5 mathematically, B2n×2n(x) = b0(E2n−2)×(2n−2)(x)) × R(2n−1)×2(x) + b1(E(2n−2)×(2n−2)(x)) × Decomposition of B2n×2n and B(2n+1)×(2n+1) R(2n−2)×2(x) + b2(E(2n−2)×(2n−2)(x)) × 3 2 P2n−1 k R(2n−4)×2(x) x + ::: + R2×(2n−k)(x) k=0 x bk(O2n×2n(x))× 2n−2 b2n−2(E(2n−2)×(2n−2)(x))×R(3×2)(x) x + R1×(2n−k+1)(x) 2n−3 b2n−3(E(2n−2)×(2n−2)(x))×R2×2(x) x where n ≥ 2. Note that bk(B(x)) refers to which is equivalent to the Qu-Sim Theo- the kth bishop coefficient of some bishop rem presented at the start of this section.