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THE q-QUEENS PROBLEM: ONE-MOVE RIDERS ON THE RECTANGULAR BOARD 5 1 JAIMAL ICHHARAM 0 2 n Abstract. We generalize the recent results of Chaiken et al. in [2] to a rectangular a mˆnchessboard. Anexplicitformulaforthenumberofnonattackingconfigurationsof J one-moveridersonsuchachessboardiscalculatedintwodifferentways,oneutilizingthe 7 theory of symmetric functions and the other the theory of generating functions. With 2 thesenewlyfoundresults,severalconjecturesandopenproblemsin[2]areresolved,and ] various formulas found byKotesovec in [4] are generalized. O C . h t 1. Introduction a m The famous n-Queens problem has had a long and storied history since its introduc- [ tion in 1850. The problem, which consists of determining the number of ways to place 1 n nonattacking queens on an nˆn chessboard, and its subsequent generalizations have v 2 eluded the attempts of many researchers to find a closed form solution and inspired the 4 development of entire branches of combinatorial mathematics, including the theory of 6 6 rook polynomials. Since the original puzzle was published, the scope of the problem has 0 . drastically expanded, with interest raised in studying the nonattacking configurations of 1 0 various different types of pieces on arbitrarily shaped boards. 5 1 : Recent work by Chaiken, Hanusa, and Zavlasky in [1] has since framed these questions v i in an algebraic perspective by relating the nonattacking configurations of pieces on a X chessboard to lattice polytopes and their associated Ehrhart quasipolynomials. In this r a work, we will combine some of their ideas along with various combinatorial techniques in order to furtherdevelop theresults of [2]on a rectangular chessboard, provingthenatural generalizations of many of the theorems found in the work and establishing some of their unresolved conjectures. In particular, we obtain two different formulas for the number of nonattacking configurations of q one-move riders on an mˆn rectangular board, and prove the conjecture found in [2] that the quasipolynomial associated with the number of such configurations on a square board has d-periodic coefficients. Date: January 28, 2015. 1 THE q-QUEENS PROBLEM: ONE-MOVE RIDERS ON THE RECTANGULAR BOARD 2 2. Two Pieces We begin by establishing the generalization of Lemma 3.1 in [2] to a rectangular board. As in the aforementioned work, we consider a move pc,dq P M with slope d{c, and define the multiset of line sizes Ld{cpm,nq :“ t|ld{cpbq| : ld{cpbq ‰ Hu, B B where ld{cpbq :“ ld{cpbqXrm,ns. B Lemma 2.1. Assume c and d are relatively prime integers such that 0 ă tnuď tmu. Let d c n¯ :“ n pmod dq. Then the multiplicities of the line sizes of Ld{cpm,nq are given in the following table: 1 ď l ă tnu :2cd tnu: pdń¯qpmćtnuq`cpn¯`dq tnu`1 : n¯pmćtnuq. d d d d d Proof. Theanalogous prooffoundin[2]holds, withmsubstitutedfornintheappropriate locations. The hypotheses of the lemma are modified from that of [2] to ensure that the analysis is correct. (cid:3) We note that by exchanging m and c with n and d if necessary, we can always satisfy the hypotheses for Lemma 2.1 regardless of the given board and slope d{c. We can therefore also calculate the number of ordered p-tuples of collinear attacking positions on rectangular boards (Proposition 3.1 of [2]). Define: αd{cpp;m,nq “ lp. lPLd{cpm,nq ÿ and notice that under the hypotheses of Lemma 2.1: n t u´1 d αd{cpp;m,nq “ 2cd lp` pdń¯qpmćtnuq`cpn¯`dq tnup`pmćtnuqptnu`1qp, d d d d l“1 ÿ ` ˘ which yields the rectangular lattice generalizations of αd{c and βd{c from [2] for p “ 2,3: 3dmn2ćn3 c n¯pdń¯q cpd´2n¯q αd{cp2;m,nq “ ` n ` dmćn´ d2 3 d2 3 ˆ ˙ ˆ ˙ 2dmn3ćn4 c αd{cp3;m,nq “ ` n2 2d3 2d ˆ ˙ n¯pdń¯q 3cn¯pdń¯q ` p3n`d´2n¯qdm´p3n`2d´4n¯qcn` . d3 2 ˆ ˙ THE q-QUEENS PROBLEM: ONE-MOVE RIDERS ON THE RECTANGULAR BOARD 3 We can therefore calculate the number of nonattacking configurations for two pieces of the same type P on a rectangular board. Theorem 2.2. For pc,dq P M, let cˆ“ minpc,dq and dˆ“ maxpc,dq, and denote by m cˆ,dˆ and n a choice of orientation such that the hypotheses of Lemma 2.1 are satisfied, i.e. cˆ,dˆ 0 ă tnu ď tmu. Then we have: dˆ c 1 |M´1| u p2;m,nq “ m2n2´ mn P 2 2 3dˆmcˆ,dˆn2cˆ,dˆćˆn3cˆ,dˆ cˆ n¯cˆ,dˆpdˆń¯cˆ,dˆq cˆpdˆ´2n¯cˆ,dˆq ´ ` n ` dm ćˆn ´ . pc,dqPM»¨ dˆ2 3 cˆ,dˆ˛ dˆ2 ˜ cˆ,dˆ cˆ,dˆ 3 ¸fi ÿ –˝ ‚ fl Proof. We firstcalculate thenumberof attacking configurations, notingthatthis canonly occur if the two pieces are along the same line of slope pc,dq for some move in M: a p2;m,nq “ αd{cp2;m ,n q´p|M|´1qmn. P cˆ,dˆ cˆ,dˆ pc,dqPM ÿ Then we can calculate the number of attacking configurations: 1 1 u p2;m,nq “ o p2;m,nq “ m2n2´ p2;m ,n q`p|M|´1qmn P P c,d c,d 2! 2» fi pc,dqPM ÿ 1 |M´1|– fl “ m2n2´ mn 2 2 3dˆmcˆ,dˆn2cˆ,dˆćˆn3cˆ,dˆ cˆ n¯cˆ,dˆpdˆń¯cˆ,dˆq cˆpdˆ´2n¯cˆ,dˆq ´ ` n ` dm ćˆn ´ . pc,dqPM»¨ dˆ2 3 cˆ,dˆ˛ dˆ2 ˜ cˆ,dˆ cˆ,dˆ 3 ¸fi ÿ –˝ ‚ (cid:3) fl 3. One-Move Riders We proceed to generalize Proposition 6.1 from [2], calculating values of u pp;m,nq for P P a one-move rider (M “ tpc,dqu with c,d relatively prime) on any convex board with line set Ld{cpBq. Proposition 3.1. For a piece P with moveset M “ tpc,dqu with 0 ď c ď d on a convex board B, k ni uPpq;Bq “ p´1qq´ ki“1ni λn1in ! lλi , n1λ1`¨¨ÿ¨`nkλk“q ř źi“1 i i ˜lPÿLd{c ¸ THE q-QUEENS PROBLEM: ONE-MOVE RIDERS ON THE RECTANGULAR BOARD 4 where the outermost sum represents a sum over all integer partitions of q, the λ denote i integers appearing in a particular partition and the n their multiplicity. i Proof. We notice that in order for q pieces on board B to be in a nonattacking config- uration, it must be the case that the q pieces are on q distinct lines l ,...,l P Ld{c. 1 q Therefore the number of nonattacking configurations is given by a sum over all possible combinations of q lines: u pq;Bq “ l l ¨¨¨l . P 1 2 q tl1,...,ÿlquĂLd{c While the above equation does indeed yield a closed-form expression for the number of nonattacking configurations, enumerating subsets of a multiset is a difficult task. Instead, using the principle of inclusion-exclusion, we obtain a weighted alternating sum over partitions of q consisting of products of sums of powers indexed by the multiset L. An elementary combinatorialargumentrelatesthecoefficientofeachsummandtotheorderof the centralizer of the permutation group with k n elements and cycle type λ ,...,λ , i“1 i 1 k which is calculated in [5]. The above expression then reads ř k ni uPpq;Bq “ p´1qq´ ki“1ni 1 1 lλi . n ! λ n1λ1`¨¨ÿ¨`nkλk“q ř źi“1 i ˜ i lPÿLd{c ¸ (cid:3) Restricting to the case of a m ˆ n rectangular board, and orienting such that the hypotheses of Lemma 2.1 are satisfied, the sum inside the parentheses can be written: tnu´1 d lλi “ 2cd lλi` dpc`mqćn`p2c´m`ctnuqn¯ tnuλi`n¯pmćtnuq tnu`1 λi, d d d d lPLd{cpm,nq l“1 ÿ ÿ ` ˘ ` ˘ by Lemma 2.1. Conjecture 6.1 from [2], which establishes the period of the coefficients of the quasipolynomial u pq;n,nq for a one-move rider on a square board, easily follows P from the above: Theorem 3.2. For a one-move rider with basic move pc,dq, the period of u pq;n,nq is P exactly maxp|c|,|d|q. Proof. It suffices to fix d ě c. By the previous discussion, we have an explicit formula for the quasipolynomial: k ni uPpq;n,nq “ p´1qq´ ki“1ni λn1in ! lλi , n1λ1`¨¨ÿ¨`nkλk“q ř źi“1 i i ˜lPÿLd{c ¸ THE q-QUEENS PROBLEM: ONE-MOVE RIDERS ON THE RECTANGULAR BOARD 5 Fixingq intheaboveexpression,weobtainapolynomialexpressionforu pq;n,nqinterms P of quasipolynomials lPLd{clλi, each of which are d-periodic. It then immediately follows that u pq;n,nq, as a polynomial of d-periodic quasipolynomials, is also d-periodic. (cid:3) P ř While the above expression is well-suited to the proof of the periodicity of the coefficients, we can obtain an alternate, more computationally efficient formula for u pq;m,nq P by noting that for a board B with multiset of lines Ld{c, the coefficient of x|L|´q in the following polynomial px`lq, lPLd{c ź yields the value of u pq;Bq. In the case of the rectangular board, the above polynomial P takes a particularly simple form: n t u´1 n n d px`tnuqpdń¯qpmćtduq`cpn¯`dqpx`tnu`1qn¯pmćtduq pxìq2cd. d d i“1 ź Calculating the generating function for the coefficients of the above polynomial therefore yields an expression for u pq;m,nq: P Theorem 3.3. Let 0ă tnu ď tmu with gcdpc,dq “ 1. For a one move rider with moveset d c M “ tpc,dqu on an mˆn rectangular board, the value of u pq;m,nq is given by P pdń¯qpmćtnuq`cpn¯`dq n¯pmćtnuq 2cd tnu d tnuj d ptnu`1qk d , j d k d tnu´x k`j`x1`ÿ¨¨¨`x2cd“qˆ ˙ ˆ ˙ źi“1„ d i where thesquarebracketsdenoteunsignedStirlingnumbersofthefirstkindandk,j,x ,...x 1 2cd are nonnegative integers. Proof. We recall the well-known result ([6], Corollary 2.4.2) that the qth coefficient of the polynomial n px`nq “ Spn,qq. Similarly, the pth coefficient of px`mqs is given by i“1 s mp. Then the above formula follows by summing over all ordered partitions of q into p ś the 2pcdtnu`1q of the above components that form the given polynomial and multiplying ` ˘ d the corresponding binomial coefficients and Stirling numbers within each summand. (cid:3) This expression can be seen as an improvement over that in Proposition 3.1, as when the one-move rider is fixed (i.e. c and d are fixed), then the outermost sum over fixed- length partitions simplifies into a finite number of sums which is easier to evaluate. In particular, the above formula can be used to rederive certain classical expressions for particular one-move riders on a rectangular board. Consider, for example, the semirook (denoted Q10 as per the convention in [3]), a piece which moves only vertically. A simple THE q-QUEENS PROBLEM: ONE-MOVE RIDERS ON THE RECTANGULAR BOARD 6 combinatorial argument gives: m uQ10pq,m,nq “ nq q ˆ ˙ To obtain this expression from the formula given in Theorem 3.3, simply consider that n¯ “ 2cd “0, and therefore the only term that survives is: pdń¯qpmćtnuq`cpn¯`dq m uQ10pq,m,nq “ qd tnduq “ q nq ˆ ˙ ˆ ˙ In a similar fashion, we can also derive the number of nonattacking configurations for a semibishop, denoted Q01. In this case, one of the binomial coeffients vanishes, and the expression simplifies to mń`1 m`n´q´l mń`1 n n uQ01pq;m,nq “ nmń´l`1 l j m`n´q´j´l`1 , l“0 ˆ ˙ j“1 „ „  ÿ ÿ where we note that m ě n. Setting m “ n, the above formula simplifies to the expression provided in [4] for semibishops on the square nˆn board. 4. Conclusion and Future Work In Proposition 3.1 and Theorem 3.3, we have found the first closed-form formulas that entirely solve the q-Queens problem on a rectangular board for a specific class of chess pieces, namely, one-move riders. This is a significant breakthrough in the course of q- Queens history, and we hope that some of the techniques utilized will be extended to make further progress on this problem. In particular, we hope to extend our results to two-move riders using the theory of rook polynomials and generating functions, although thistaskappearstobemuchmoredifficultduetoourinabilitytoobtainclosedexpressions for rook polynomials for the necessary classes of skew Ferrers boards. References [1] Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-queens problem. I. General Theory. Electronic Journal of Combinatorics, Volume21, Issue3, Aug.2014; [2] Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-queens problem. II. The squareboard. Journal of Algebraic Combinatorics, Aug.2014; [3] Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-queens problem. III. Partial Queens. arXiv 1402.4880. [4] VaclavKotesovec,Non-attackingchesspieces(chessandmathematics).[Self-publishedonlinebook], 6th ed. Feb. 2013, URL:http://web.telecom.cz/vaclav.kotesovec/math.htm THE q-QUEENS PROBLEM: ONE-MOVE RIDERS ON THE RECTANGULAR BOARD 7 [5] I. G. MacDonald, Symmetric Functions and Hall Polynomials, 2nd ed. Oxford University Press, 1995. [6] RichardP. Stanley,Enumerative Combinatorics,Vol.1,2nded.CambridgeUniversityPress,2012. Harvard University, Cambridge, MA 02138, USA. E-mail address: [email protected]