RELAXED COMPLETE PARTITIONS: AN ERROR-CORRECTING BACHET S PROBLEM

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1 #A4 INTEGERS 8 (208) RELAXED COMPLETE PARTITIONS: AN ERROR-CORRECTING BACHET S PROBLEM Joge Buno Depatment of Accounting, Finance, Mathematics and Economics, Univesity of Wincheste, Wincheste, United Kingdom Joge.Buno@wincheste.ac.uk Edwin O Shea Depatment of Mathematics and Statistics, James Madison Univesity, Haisonbug, Viginia osheaem@jmu.edu Received: 4/3/4, Revised: /7/8, Accepted: 4/8/8, Published: 5/8/8 Abstact Motivated by an eo-coecting genealization of Bachet s weights poblem, we define and classify elaxed complete patitions. We show that these patitions enjoy a succinct desciption in tems of lattice points in polyheda. Ou main esult is an enumeation of the minimal such patitions (those with fewest possible pats) via Bion s fomula. This genealizes wok of Pak on classifying complete patitions and that of Rødseth on enumeating minimal complete patitions.. Intoduction Bachet s weights poblem asks fo the least numbe of intege weights that can be used on a two-pan scale to weigh any intege between and 40 inclusive. Its unique solution consists of fou pats and can be witten as the intege patition 40 = While the poblem was populaized by Bachet in 62 [2, page 43], its noble oots can be taced to Fibonacci s Libe Abaci [6, On IIII Weights Weighing Foty Pounds]. Its naming as Bachet s Weights Poblem occus in numeous souces, including influential texts fom the ealy twentieth centuy like Hady and Wight s Theoy of Numbes [0, 9.7] and Ball s popula Mathematical Receations and Essays [3, Chapte ]. Despite these authos collective awaeness that Bachet only ecoded the poblem Ball cedits Tataglia fo solving the weights poblem in 556 the attibution to Bachet has endued. Genealizations of Bachet s poblem eplacing 40 with any intege include MacMahon s pefect and subpefect patitions [], Bown s complete patitions [8] JB was suppoted by an EMBARK fellowship of the Iish Reseach Council fo Science, Engineeing & Technology.

2 INTEGERS: 8 (208) 2 and Pak s extension of Bown s wok [5]. These genealizations begin with the simple obsevation that a patition m = n ealizes evey intege between m and m on a two-scale pan if and only if { P n i=0 i i : i 2 {0,, 2}} = {0,, 2,..., 2m}; the solution to the oiginal poblem now follows fom the unique base 3 epesentation of evey intege. Tanton [9] consideed an eo-coecting vaiant of Bachet s poblem. Given a fixed but unknown intege weight l, weighing no moe than 80 pounds, what is the least numbe of intege weights that can be used on a two-pan scale to discen l s value? We still need only fou pats, the patition 80 = su ces: if the unknown intege weight l is even then its exact value will be ealized by the pats of with a balanced scale, if the weight l is odd then both l and l + can be achieved by the pats of and the unknown weight will be heavie than the fome and lighte than the latte. Equivalently, the pats of 80 = can be used to weigh evey intege between and 80 on a two-scale pan allowing fo an eo of one. Tanton s vaiant along with the simple obsevation common to pevious genealizations suggests the following definition. Definition. A patition m = n with the pats in weakly inceasing ode is an e-elaxed -complete patition ((e, )-patition fo shot) if no e + consecutive integes between 0 and m ae absent fom the set { P n i=0 i i : i 2 {0,,..., }}. An (e, )-patition with the fewest numbe of pats is called minimal. Pak s -complete patitions [5] ae the (0, )-patitions. The (0, )-patitions ae the vaiant of Bachet s poblem whee weights ae allowed on only one of the two pans and wee fist intoduced as Bown s complete patitions [8]. MacMahon s pefect and subpefect patitions [] ae the (0, )-patitions and (0, 2)-patitions espectively, but with additional conditions that genealize the uniqueness of epesentations in a given base. Tanton s vesion of Bachet asks fo the minimal (, 2)- patitions of 80. The (0, )-complete patitions wee enumeated by Pak [5]. The minimal patitions wee enumeated by O Shea [3] (a patial enumeation fo the = case) and by Rødseth ( = [7] and 2 [8]). We povide a complete classification of minimal (e, )-patitions as sets of intege points in polyheda (Theoem and Poposition in Section 2). These sets ae enumeated using Bion s theoem [7], a fomula that encodes lattice points in a given polyhedon by fist attaching a geneating function to each vetex in the polyhedon and then taking the fomal sum of these geneating functions. In Section 3, we attain ou main esult by applying Bion s theoem to P n (e, ), a tansfomed vesion of the polyhedon of the minimal (e, )-patitions with n + pats: Theoem 2. If 2, the minimal elaxed complete patitions ae enumeated by nx the geneating function Pn(e,)(x) = Kµ(;) (x) + K µ({j}) (x)+o(x (e+)(+)n ). j=

3 INTEGERS: 8 (208) 3 The polyhedon P n (e, ) is combinatoially isomophic to the (n + )-cube. Theoem 2 states that we only need enumeate n + of P n (e, ) s 2 n+ vetex cones the vetex cone at the oigin, indexed by K µ(;), and all but one of the vetex cones of those vetices neighboing the oigin, those indexed by K µ({j}). Each of these vetex cones and thei geneating functions ae descibed explicitly in Section 3. Theoem 3 addesses the = case. Ou esults agee with those of Pak, O Shea, and Rødseth when e = 0. This use of discete geometic tools to enumeate families of intege patitions descibed by linea inequalities contibutes to an existing liteatue that is especially inspied by the 997 papes [5, 6] of Bousquet-Mélou and Eiksson on lectue hall patitions. These patitions, aising fom Bott s fomula fo enumeating the lengths of educed wods in the a ne Coxete goups of type C, can be descibed in tems of lattice points in polyheda, and they povide a efinement of Eule s classical odd-distinct theoem. Othe papes in the same discete geometic spiit include Pak s use of lattice point enumeation when the family of intege patitions ae points in a unimodula cone [4], Andew s poof of the lectue hall theoem [] utilizing MacMahon s Patition Analysis (the opeato) [2, Vol. 2, VIII], and Coteel, Lee, and Savage s boad intoduction to discete geometic methods in intege patitions [9]. What the (e, )-patitions shae with lectue hall patitions is that they both genealize classical intege patition poblems by viewing them though a moden geometic lens. To illustate ou main esult, conside the six minimal (, 2)-patitions of m = 2: +3+8, +4+7, 2+2+8, 2+3+7, and These ae tansfomed to six µ lattice points via µ i := (e+)( +) i i: (, 3, 0), (, 2, ), (0, 4, 0), (0, 3, ), (0, 2, 2) and (0,, 3) espectively. These six lattice points have coodinate sum equal to 26 2 = 4 in the polyhedon P 2 (, 2) below. By consulting Figue and Theoem 2, ou enumeating function fo the lattice

4 INTEGERS: 8 (208) 4 Figue : The polyhedon P 2 (, 2) with vetices and vetex cones labeled. points in P 2 (, 2) is: P 2(,2)(x) = Kµ(;) (x) + Kµ({}) (x) + Kµ({2}) (x) + O(x 8 ) = x (0,0,0) x (,2,6) x (0,,2) x (0,0,) = + x (0,4,8) x (0,, 2) x (,,4) x (0,0,) + x (0,0,2) x (0,0, ) x (,2,2) x (0,,) + O(x8 ) ( x 9 )( x 3 )( x ) x 3 ( x)( x 5 )( x 2 ) + O(x8 ). x 5 ( x 3 )( x 6 )( x ) The coe cient of x 26 2 = x 4 equals 7 0 = 6 as expected. 2. Classifying Relaxed Complete Patitions In this section we classify the (e, )-patitions as a collection of lattice points in a polyhedon and descibe the minimal such patitions. Call { P n j=0 j j : j 2 {0,,..., }} the -cove of n.

5 INTEGERS: 8 (208) 5 Theoem. A patition m = n such that 0 apple e+ is an (e, )-patition if and only if i apple (e + ) + P i j=0 j fo all i apple n. Poof. If i > (e + ) + P i j=0 j then the shifted set P i j=0 j + {, 2,..., e + } is omitted fom the -cove of n. We show necessity by induction on the numbe of pats in the patition. If n = 0 then 0 = m apple e + in accodance with ou hypothesis. Let n be an (e, )-patition onto which we append any pat n with n apple n apple (e + ) + P n j=0 j. We wish to show that evey positive intege l apple ( n ) is within a distance of at most e+ of some intege in the -cove of n. If l apple ( n ), then the induction hypothesis eadily applies. So we can fix l in the inteval P n j=0 j < l apple P n j=0 j. In this case thee will always exist an n, apple n apple, such that ( n ) n + P n j=0 j < l apple n n + P n j=0 j, o n = l P n j=0 j n. Since l n n apple P n j=0 j, the inductive hypothesis implies that l n n is within distance e + of an intege in the -cove of n and so l must be within distance e + of an intege in the -cove of n. Pak [5] gave a simila poof of the above theoem fo e = 0. A novel featue of Theoem is that the defining hypeplanes of the polyheda that cut out the (e, )-patitions aise fom tanslating by e the defining hypeplanes of the (0, )- patitions. Some simple coollaies include: If n is a (0, )-patition of m then (e+) 0 +(e+) + +(e+) n is an (e, )-patition of (e+)m. The patition m = ++ + is always an (e, )-patition of m. Evey (e, )-patition of m is both an (e +, )-patition and an (e, + )-patition of m. The minimal (e, )-patitions ae those (e, )-patitions with fewest possible pats. Using the inequalities of Theoem, a standad inductive agument shows that i apple (e + )( + ) i () fo all (e, )-patitions m = n. The size of any such patition m cannot exceed P n i=0 (e + )( + )i = (e+) (( + ) n+ ). That is, m apple e + (( + ) n+ ) < e + m ( + ) n+, o log + < n +. e + Since n+ is an intege, the intege pat of log + ( m e+ ) is stictly less than n+, o blog + ( m e+ )c apple n, so an (e, )-patition of m must have at least blog +( m e+ )c + pats. The constuction of an (e, )-patition with this numbe of necessay pats would ensue that this su ces fo the numbe of pats of a minimal such patition. Poposition. A minimal (e, )-patition of m has blog + ( m e+ )c + pats.

6 INTEGERS: 8 (208) 6 Poof. We claim that, fo any intege m, the patition whose pats ae exactly those in the multiset { (e+), (e+)(+), (e+)(+) 2,..., (e+)(+) n, m (e + ) ((+) n )} is an (e, )-patition of m. Since +( +)+( +) 2 + +( +) n is a (0, )-patition of (( +)n ), it follows that (e + ) + (e + )( + ) + (e + )( + ) (e + )( + ) n is an (e, )-patition of (e+) (( + ) n ). Fo each 0 apple apple, the shifted set ( n ) (e + ) X m (( + ) n ) + i (e + )( + ) i : i 2 {0,,..., } i=0 will not omit any consecutive (e + ) integes and, since m < e+ ( + )n+, the union ove apple apple of these shifted sets anges fom to m and does not omit any (e + ) consecutive integes. This union is pecisely the -cove of the patition whose pats consist of the elements fom the given multiset. The above poposition also tells us that minimality is peseved by the multiplication of pats by e +. This is consistent with the doubling of pats in solving Tanton s vaiant of Bachet s poblem with 80 = fom 40 = In summay, the minimal (e, )-patitions ae ealized as ( 0,..., n) 2 Z n+, each lying inside the polyhedon defined by the inequalities i apple i apple (e + ) + P i j=0 j fo each i apple n. The emainde of this aticle focuses on the geometic chaacte of the patitions stating with a desciption of Bion s fomula [7]. In the inteests of bevity, we will give a desciption of Bion s fomula in a manne best suited to ou needs. Given an intege vecto u define the pimitive pat of u as p(u) := gcd(u) u, whee the gcd(u) is the geatest common diviso of the enties of u. Call u pimitive if gcd(u) equals. Given two integal vectos u and u 0 wite p(u 0 u) fo p u (u 0 ), the pimitive vecto of u 0 elative to u. Note that p u (u 0 ) = p u 0(u). Given a vetex v of a polyhedon P, the vetex cone of P at v is the smallest cone with apex v that contains P. If the polyhedon P is integal (its vetices ae lattice points), letting N (v) denote the neighbos of v in P, we can wite the vetex cone at v as K v := v + R 0 {p v (v 0 ) : v 0 2 N (v)} whee the latte component is undestood as the non-negative eal span of the set of pimitive vectos of the neighbos of v elative to v itself. The polyhedon P is said to be simple if the geneatos {p v (v 0 ) : v 0 2 N (v)} of each vetex cone K v fom a linealy independent set. Assuming that the polyhedon P is full dimensional, we say that a vetex cone K v is unimodula if the squae matix with ow vectos equal to {p v (v 0 ) : v 0 2 N (v)}

7 INTEGERS: 8 (208) 7 has deteminant ±. The vetex cone K v being unimodula implies that evey lattice point in K v can be witten uniquely as the apex v plus a non-negative intege combination of the pimitive vectos that geneate the cone. Consequently, the set of intege points in a unimodula vetex cone K v can be witten as a geneating function X Y K v (z) := z m = z v z, p m2k v\z n+ v 0 v (v0 ) 2N (v) whee u = (u 0, u,..., u n ) 2 K v is encoded as z u := z u0 0 zu zun n. The fomula of Bion [7], specialized hee fo a simple, intege polyhedon P with unimodula vetex cones, states that the lattice points in P ae encoded pecisely by the monomials appeaing in the sum of the geneating functions fo the vetex cones: P(z) = X v a vetex of P K v (z) = X v a vetex of P z v Y v 0 2N (v) z p v (v0 ). (2) Bion s fomula holds in geate geneality than pesented hee. The polyhedon need not be simple, no integal, no must all vetex cones be unimodula. These geneal cases equie a moe detailed desciption [4, Chapte 9] which we do not need to call upon hee. To enumeate all lattice points with common -nom (denoted by ), it su ces to set the vaiables z 0, z,..., z n in (2) to a common vaiable x to yield P(x) = X v K v (x) = X v x v Y v 0 2N (v) x p v (v0 ), (3) whee the sum is taken ove all v that ae vetices of P. Note that if p v (v 0 ) < 0 x p v then = 0 (v) and x p v (v0 ) x p v 0 (v) K v (x) = x v Y v 0 : p v (v 0 ) <0 x p v 0 (v) x p v 0 (v) Y v 0 : p v (v 0 ) >0 x p v (v0 ), (4) whee each poduct is taken ove v 0 2 N (v). Witten in this fom, evey monomial in the vetex cone s geneating function Kv (x) has degee at least X O(K v ) := v + p v 0(v) v 0 : p v (v 0 ) <0 and we call this quantity the ode of the vetex cone. We will sometimes wite O(v) fo the ode of the vetex cone with apex v in P.

8 INTEGERS: 8 (208) 8 3. Enumeating Minimal Relaxed Complete Patitions We have established that the minimal complete patitions with n + pats ae lattice points in a polyhedon sitting in R n+. We will show that Bion s fomula, with the pemises peviously stated, applies to this polyhedon and then poceed to compute the geneating functions of the polyhedon s vetex cones. Recall that the (e, )-patitions with n + pats ae pecisely the positive intege points = ( 0,,..., n) that satisfy the linea inequalities i apple i and ineq i : i apple (e + ) + P i j=0 j, fo each apple i apple n. Futhemoe, the patitions with size := n lying in the inteval [(e + ) (+)n +, (e + ) (+)n+ ] ae the minimal such patitions. Echoing Rødseth s enumeation of the minimal (0, )patitions [7, 8], conside the following tansfomation of a patition : µ i := (e + )( + ) i i fo each i = 0,,..., n. The constaints on tanslate to constaints on µ: Xi 0 apple µ 0 apple e and µ j apple µ i apple (e + )()( + ) i + µ i. j=0 Thee is a one-to-one coespondence between the s and the µ s and since µ = e+ (( + )n+ ), the (e, )-patitions with 2 [(e + ) (+)n +, (e + ) (+)n+ ] coespond to the lattice points µ = (µ 0,..., µ n ) that satisfy the above inequalities with nom constaint µ 2 [0, (e + )( + ) n ]. Hence, it will su ce to find a geneating function fo the noms of the tansfomed µ points, P x µ + O(x (e+)(+)n ). The lattice points µ = (µ 0, µ,..., µ n ) ae contained in the half-spaces min + (0) := {µ 2 R n+ : 0 apple µ 0 }, max + (0) := {µ 2 R n+ : µ 0 apple e}, and, fo each i =, 2,..., n, Xi min + (i) := {µ 2 R n+ : µ j apple µ i } and j=0 max + (i) := {µ 2 R n+ : µ i apple (e + )()( + ) i + µ i }. Fo evey 0 apple i apple n, let min(i) and max(i) denote the defining hypeplanes of the above half-spaces, with evey inequality eplaced by equality in min + (i) and max + (i) espectively. Define the polyhedon P n (e, ) as the common intesection

9 INTEGERS: 8 (208) 9 of the above µ-half-spaces 2, P n (e, ) := n\ min + (i) \ max + (i). i=0 Fo each S {0,, 2,..., n}, define the set of points µ(s) := {µ 2 max(i) : i 2 S} \ {µ 2 min(i) : i /2 S} R n+. Poposition 2. The polyhedon P n (e, ) is an (n + )-dimensional simple polyhedon. Each µ(s) is a singleton and the vetices of P n (e, ) ae {µ(s) : S {0,, 2,..., n}}. Two vetices µ(s) and µ(t ) shae an edge if and only if thei symmetic di eence S4T is a singleton. Poof. It is staightfowad to check that thee is no point µ 2 P n (e, ) that simultaneously satisfies both max(i) and min(i). Since each min/max(i) sequentially intoduces a new independent vaiable µ i, the nomals to the hypeplanes that define µ(s) ae a set of linealy independent vectos and each µ(s) equals the intesection of n + a nely independent hypeplanes in R n+. While no othe defining hypeplane of P n (e, ) can contain µ(s), it is nonetheless contained in the inteio of the othe defining half-spaces of P n (e, ). This implies that each µ(s) is a singleton and, futhe, its single element must be a vetex of P n (e, ). Since the vetex µ(s) is defined by n+ hypeplanes (and no othes), it must have n+ edges in P n (e, ). That is, P n (e, ) is an (n + )-dimensional simple polyhedon. As fo the edges, S4T = if and only if µ(s) and µ(t ) have pecisely n of thei n + defining hypeplanes in common. Since P n (e, ) is simple, this is equivalent to µ(s) and µ(t ) being neighbos in P n (e, ). Lemma. The vetex cones of P n (e, ) ae unimodula. Poof. By Poposition 2, it su ces to study pais of sets S and T whee S4T = {j}. Fo now, let us assume that T = S [ {j} and we will show that p := p µ(s) (µ(t )) equals 0 in enties 0 though j and in enty j. Fo evey i < j, since the sets S and T agee when esticted to the suppot of {0,, 2,..., j }, µ(s) j = µ(t ) i and p µ(s) (µ(t )) i = 0. The j-th enty of the di eence vecto is µ(t ) j µ(s) j = (e + )( + ) j + µ(t ) j ( P j k=0 µ(s) k), which is a positive intege this di eence must be positive since it equals ( j ) j in the oiginal -fomulation of the (e, )-patition. 2 When e = 0, min + (0) \ max + (0) equals the hypeplane defined by µ 0 = 0. The claims of this section assume that e but when e = 0 the statements equie only mino alteation, if any. Fo example, the polyhedon P n(0, ) is of dimension n as opposed to n +.

10 INTEGERS: 8 (208) 0 Fo i = j +,..., n, ecalling that S and T agee outside of {j}, the i-th enty of the di eence equals 8 9 µ(t ) i µ(s) i : fo i 2 T >< >= µ(t ) i µ(s) i = Xi. (5) >: µ(t ) k µ(s) k : fo i /2 T >; k=j Each µ(t ) i µ(s) i is ecusively defined fom {µ(t ) k µ(s) k : k = j,..., i } and each such di eence is a multiple of µ(t ) j µ(s) j. Hence, µ(t ) j µ(s) j equals gcd(µ(t ) µ(s)) and so p µ(s) (µ(t )) j =. If S = T [ {j} then p µ(t ) (µ(s)) = p µ(s) (µ(t )) and this case follows in the same fashion except fo in enty j. Consequently, the pimitive vectos of the vetex cone of any given vetex of P n (e, ) fom the ows of a squae matix with deteminant ±. Applying Bion s fomula (Equation (4)) equies explicit calculation of the noms of the vetices of P n (e, ) and thei vetex cones. Since we ae only inteested in the lattice points of P n (e, ) that have ode less than (e + )( + ) n, we can eliminate all vetex cones with ode geate than o equal to (e + )( + ) n. To calculate µ(s) fo each S {0,, 2,..., n}, a peliminay lemma is equied. Witing µ as shothand fo µ(s), µ k0 k fo µ k + µ k+ + + µ k 0, µ k0 fo µ k0 0, and µ k fo µ n k, we make the following claims. Lemma 2. (a) If k,..., k 0 µ k0 k = (( + ) k0 k (b) If k /2 S then µ k = (c) If 0 < k,..., k 0 µ k0 k = + µ k. 2 S then " (e + )( + )k /2 S then ) µ k and µ k0 = ( + ) k0 k µ k. ( + ) k0 k (d) If 0,..., k 0 2 S then µ k0 0 = # (k 0 k) + (k 0 k)µ k. + " # (e + ) ( + ) k0 k 0 e +. Poof. (a) Since k /2 S, it follows that µ k = µ k. Similaly, k + /2 S implies µ k+ = µ k = ( µ k + µ k ) = ( µ k + µ k ) = ( + ) µ k. Recusively, µ k+i = ( + ) i µ k, hence µ k0 k k 0 X k = ( + ) i µ k = (( + ) k0 k i=0 ) µ k.

11 INTEGERS: 8 (208) It follows that µ k0 = µ k0 k + µ k. (b) Since µ j = µ j + µ j and j /2 S, we have µ j = µ j. That is, µ j = ( + )µ j. (c) Since k 2 S, we have that µ k = (e + )( + ) k + µ k. Similaly, k + 2 S implies µ k+ = (e + )( + ) k + µ k = (e + )( + ) k [( + ) + ] + µ k. Recusively, fo evey i = 0,,..., (k 0 ) k, we have µ k+i = (e + )( + ) k [( + ) i + ( + ) i + + ( + ) + ] + µ k = (e + )( + ) k [( + ) i+ ] + µ k and so µ k0 k = (e + )( + ) k P k 0 k i=0 [( + ) j+ ] + P k 0 k i=0 µ k, o " # µ k0 k = (e + )( + ) k ( + ) (( + ) k0 and is easily ewitten in the desied fom. k ) (k 0 k) + (k 0 k)µ k (d) Follows in the same manne as (c), with the initial condition of µ 0 = e. With Lemma 2 in hand, we can compute the nom of each vetex of P n (e, ). If ( i, n i ) := { i, i +,..., i + n i } S then ( i, n i ) is said to be a chain in S; it is said to be maximal if no othe chain in S popely contains ( i, n i ). We can wite a set S in tems of the union of its maximal chains vis-a-vis S = {(, n ), ( 2, n 2 ),..., ( p, n p )}. Since each chain is maximal, it follows that i + n i < i+ and S = n + n n p. The following esult descibes the nom of any vetex µ(s) in tems of the maximal chains in S. Lemma 3. Fo S = {(, n ),..., ( p, n p )} with > 0, " (e + )( + )n+ S py µ(s) = ( + ) S + n # i. (6) + Fo S = {(0, n ),..., ( p, n p )}, (e + )( + )n+ S µ(s) = "( + ) S + n py + n # i. (7) e + + Poof. We pove the > 0 case by induction on the numbe of maximal chains in S, leaving the eade to confim the = 0 case in a simila fashion. Fo the base case of S = {(, n )}, setting k = + n and k 0 = n in Lemma 2(a), the vetex µ has nom µ = (+) n+ n µ +n. The quantity i= i=2

12 INTEGERS: 8 (208) 2 µ +n is computed by setting k = and k 0 = + n in Lemma 2(c): µ = ( + ) n+ n µ +n " # n)+ (e + )( + )(n+ = ( + ) n (n ) + (n )0. + Next, assume the claim is tue fo any set S 0 with at most p chains. Let S = {(, n ),..., ( p, n p )} be a subset of {0,, 2,..., n} with p chains. Since p + n p,..., n 62 S, it follows fom Lemma 2(a), that µ(s) = µ p+np + µ n p+n p = ( + ) n+ ( p+np) µ p+np = ( + ) n+ ( p+np) ( µ p + µ p+np p ). Lemma 2(c) with k = p and k 0 = p + n p applies to µ p+np p to yield µ p+np p = (e + )( + ) p+np (e + )( + ) p n p (e + )( + ) p + n p + µ p. It follows that µ p+np equals µ p + µ p+np p = (e+)(+) p+np (e+)(+) p n p (e + )( + ) p + ( + n p + ) µ p. (8) Setting S 0 = {(, n ),..., ( p, n p )} as a subset of {0,, 2,..., p }, the inductive hypothesis allows fo µ p = (e + )( + ) p (e + )( + ) p S0 py + n i. (9) + Substituting the expession fo µ p in (9) into the last tem of (8) gives µ p + µ p+np p i= = (e + )( + ) p+np (e + )( + ) p n p (e + )( + ) p (e + )( + ) p + + n p + (e + )( + ) p S 0 py + n p + n i. + + i=

13 INTEGERS: 8 (208) 3 The second and thid tems cancel with the fouth so µ p+np = (e + )( + ) p+np (e + )( + ) p S0 + n p + and that last poduct equals Q p i= + ni +. Finally, µ = ( + ) n+ p np ( µ p + µ p+np p ) = (e + )( + )n+ (e + )( + ) n+ S0 n p p as claimed. The case whee = 0 is poved in the same manne. Y i= py i= + n i, + + n i + Given an abitay S = {(, n ),..., ( p, n p )} and a choice of = 0 o > 0, the pevious lemma makes no efeence to the values of the emaining i s so we can explicitly calculate the noms of all vetices of P n (e, ). Coollay. (a) If S = {j} then (e)( + ) µ = n : j = 0 (e + )( + ) n : j > 0. (b) If S = {j, j 2 } then 8 >< µ = >: (e + )( + ) n + [e ] ( + ) n : 0 = j = j 2 (e + )( + ) n + [(e ) )] ( + ) n 2 : 0 = j < j 2 (e + )( + ) n + [ ] (e + )( + ) n 2 : 0 < j = j 2 (e + )( + ) n + [( ) ] (e + )( + ) n 3 : 0 < j < j 2 (c) If S = {j, j 2, j 3 } with = then 8 < µ = : (e + )2 n + (7e 2)2 n 4 : 0 = j < j 2 < j 3 2 (e + )2 n + (5e )2 n 3 : 0 = j = j 2 < j 3 2 (e + )2 n + 7(e + )2 n 5 : 0 < j < j 2 < j 3 2 Poof. Pat (b) when 0 < j < j 2 is a special case of Lemma 3 with S = {(, n ), ( 2, n 2 )}, > 0 and n = n 2 = : # 2 2 (e + )( + )n+ µ = "( + ) 2 + ( + ) = (e + )( + ) n 3 ( + ) 3 + ( ) = (e + )( + ) n + [( ) ](e + )( + ) n 3. The othe claims follows in a simila fashion fom Lemma 3. 9 = ;. 9 >=. >;

14 INTEGERS: 8 (208) 4 With the noms of the vetices µ(s) of P n (e, ) with S apple 2 computed, we tun to the pimitive vectos associated to those vetices. A coollay of Equation (5) (with the geatest common diviso tem emoved) is that we can descibe pimitive vectos ecusively. If T = S [ {j} then 8 0 : i < j >< : i = j p µ(s) (µ(t )) i = p µ(s) (µ(t )) i : i > j, i 2 T >: P i k=j p µ(s)(µ(t )) k : i > j, i /2 T This allows us to explicitly compute the noms of the pimitive vectos. 9 >=. (0) >; Coollay 2. When S apple 2, the noms of the pimitive vectos associated with the vetex µ(s) ae as follows: (a) p µ({;}) (µ({j})) = ( + ) n j fo each j = 0,,..., n. () (b) If µ(s) = {k} and T = {j, j 2 }, whee j < j 2 : 8 < p µ({s}) (µ(t )) = : ( + ) n j2 : k = j 2( + ) n j : k = j 2 and j = j 2 (2 + )( + ) n j 2 : k = j 2 and j < j 2 9 = ;. (2) (c) If S = {j, j 2 } with j < j 2 and T = {j, j 2, k} then 8 9 (2 + ) 2 ( + ) n k 4 : k < j >< 2(2 + )( + ) n k 3 : k = j >= p µ(s) (µ(t )) = (2 + )( + ) n k 2 : k < j 2. (3) 2( + ) >: n k : k = j 2 >; ( + ) n k : k > j 2 Poof. In each case, the vectos µ(t ) µ(s) can be constucted ecusively fom Equation (0), and the claims follow fom the geometic seies + ( + ( + ) + + ( + ) k ) = ( + ) k. Fo example, 8 9 < 0 : i < j = p µ(;) (µ({j})) i = : i = j : ( + ) i j ; : i > j and so p µ(;) (µ({j})) = + ( ( + ) n j ) = ( + ) n j. We ae now on the cusp of poving ou main esult, the enumeation of the minimal elaxed complete patitions. Recall that the ode of the vetex cone at µ(s) is O(µ(S)) = µ(s) + P T p µ(t )(µ(s)), whee the sum is taken ove each T with T = S {j}, and that we ae only inteested in those vetex cones with ode less than (e + )( + ) n.

15 INTEGERS: 8 (208) 5 Lemma 4. If S T then µ(s) apple µ(t ). Poof. As noted in Lemma, each µ(t ) j µ(s) j 0, with stict inequality when j 2 T S. Lemma 5. If 0 2 S and S apple 2 then O(µ) (e + )( + ) n. Poof. All claims egading µ and the noms of pimitive vectos can be ead diectly fom Coollaies and 2 espectively. Fo S = {0}, µ = e( + ) n and p µ(;) (µ({0})) = ( + ) n, which implies O(µ({0})) µ({0}) + p µ(;) (µ({0})) = e( + ) n + ( + ) n = (e + )( + ) n. Fo S = {0, }, µ = (e + )( + ) n + [e ]( + ) n and p µ({0}) (µ({0, })) = ( + ) n, which implies O(µ) (e + )( + ) n + [e ]( + ) n + ( + ) n (e + )( + ) n. Fo S = {0, j 2 } with j 2 >, µ = (e + )( + ) n [(e ) ]( + ) n 2 and p µ({j2})(µ({0, j 2 })) = (2 + )( + ) n 0 2, which implies O(µ) (e + )( + ) n + [(e ) ]( + ) n 2 + (2 + )( + ) n 2 (e + )( + ) n. Theoem 2. If 2 then Pn(e,)(x) = Kµ(;) + nx j= K µ({j}) + O(x (e+)(+)n ). Poof. We claim that O(µ) (e + )( + ) n wheneve (i) S 2, o (ii) 0 2 S. (i): If S = {j, j 2 } with 0 < j < j 2 then, by Coollay (b), µ(s) (e+)(+) n. Any set T of size 3 o geate contains a set of the fom S = {j, j 2 } and, by Lemma 4, µ(t ) µ(s) (e + )( + ) n. (ii): If 0 2 S with S 3 then S would contain a set of the fom {j, j 2 } with j > 0 so, by case (i) and Lemma 4, µ (e + )( + ) n. By Lemma 5, if S = o 2 then O(µ) (e + )( + ) n. Hence O(µ(S)) (e + )( + ) n fo evey set S that contains 0. The = case follows in much the same way but thee ae exta tems in the geneating fuction. Theoem 3. We have nx P n(e,)(x) = Kµ(;) + j= K µ({j}) + nx nx j = j 2=j +2 K µ({j,j 2 }) + O(x(e+)2n ). Poof. We claim that O(µ) (e + )2 n wheneve (i) S = {j, j + }, (ii) S 3, o (iii) 0 2 S. (i): If S = {j, j 2 } with 0 < j = j 2 then, by Coollay (b), µ (e + )2 n. Applying Lemma 4, any T with two consecutive non-zeo elements has µ(t )

16 INTEGERS: 8 (208) 6 (e + )2 n. In contast to 2, when = and S = {j, j 2 } with 0 < j < j 2, µ = (e + )2 n + [2(2 ) ] (e + )2 n 3 = (e + )2 n (e + )2 n 3 < (e + )2 n. Using Coollay 2, one can check that O(µ) < (e + )2 n, hence S = {j, j 2 } with 0 < j < j 2 appeas as the index fo the double summation in the = geneating function. (ii): If S 3 and S contains two consecutive elements then, by (i) and Lemma 4, µ (e + )2 n. Altenatively, if S does not contain two such elements (i.e., S = {j, j 2, j 3 } with 0 < j < j 2 < j 3 2) then, by Coollay (c), µ(s) µ({j, j 2, j 3 }) (e + )2 n. (iii): Assume 0 2 S. If S apple 2 the claim follows diectly fom Lemma 5. If S 4 then S must contain thee non-zeo elements and case (ii) combined with Lemma 4 ensues that µ(s) (e + )2 n. All that emains ae the thee element sets S = {0, j 2, j 3 } with j 2 < j 3. If S = {0,, j 3 } then µ = (e + )2 n + [5e ]2 n 3 and p µ({,j3})(µ(s)) = 2(3)2 n 3, so O(µ) (e+)2 n +[5e +6]2 n 3 = (e+)2 n +5(e+)2 n 3 (e+)2 n. Othewise, S = {0, j 2, j 3 } with j 2 > then µ({0, j, j 2 }) = (e+)2 n +(7e 2)2 n 4 and p µ({j2,j 3})(µ(S)) = n 4, so O(µ) (e+)2 n +7(e+)2 n 4 (e+)2 n. We close with explicit desciptions of the above geneating functions. Reading diectly fom Equation (4) and using Coollaies and 2 we have: K µ(;) = x 0 n Y j=0 x p µ(;) (µ({j}) = ny j=0 x (+)j = and K µ({j }) = x(e+)(+)n x p µ(;) (µ({j}) x p µ(;) (µ({j}) Y = x (e+)(+)n x(+)n j ny k=j + x (e+)(+)n +(+) n j n j ( x (+)n j )( x 2(+)n j ) Y t=0 k6=j x p µ({j }) (µ({j,k}) x (+)n j x 2(+)n j jy 2 x (+)n k x (2+)(+)n k 2 k=0 x (+)t ny 2 t=n j x (2+)(+)t.

17 INTEGERS: 8 (208) 7 Fo S = {j, j 2 } with 0 < j < j 2 and =, K µ({j,j 2 }) = x(e+)7 2n 3 Y k2{j,j 2} Y k62{j,j 2} x p µ({k}) (µ({j,j2}) x p µ({k}) (µ({j,j2}) x p µ({j,j 2 }) (µ({j,j2,k}) n Yj 2+ = x (e+)7 2n 3 x 2n j 2 x 3 2n j 2 n Yj x 3 2t t=n j 2 ny 4 t=0 t=n j 2 x 2t x 9 2t. Setting (e, ) = (, 2) and n = 2 we ecove the enumeating function P2(,2)(x) descibed in the intoduction. Setting e = 0 povides Rødseth s enumeating functions [7, 8] fo the minimal complete patitions. Refeences [] G.E. Andews MacMahon s patition analysis. I. The lectue hall patition theoem. In Mathematical essays in hono of Gian-Calo Rota (Cambidge, MA, 996), volume 6 of Pog. Math., pages -22. Bikhäuse Boston, Boston, MA, 998. [2] C.G. Bachet de Méziiac, Poblèmes Plaisans et Delectables, qui se font pa les Nombes, Piee Rigaud, 62. [3] W.W.R. Ball, Mathematical Receations and Essays, (Fouth Edition), Macmillan, 905. [4] M. Beck and S. Robins Computing the Continuous Discetely: Intege-Point Enumeation in Polyheda, Spinge, [5] M. Bousquet-Mélou and K. Eiksson, Lectue hall patitions, Ramanujan J. (997), 0-. [6] M. Bousquet-Mélou and K. Eiksson, Lectue hall patitions II, Ramanujan J. (997), [7] M. Bion, Points enties dans les polyèdes convexes, Ann. Sci. cole Nom. Sup. 2 (988), [8] J.L. Bown, Note on complete sequences of integes, Ame. Math. Monthly 68 (96), [9] S. Coteel, S. Lee, and C. Savage, Five guidelines fo patition analysis with applications to lectue hall-type theoems, in Combinatoial numbe theoy, de Guyte, Belin, [0] G.H. Hady and E.M. Wight, An Intoduction to the Theoy of Numbes, (Sixth Edition), Oxfod Univesity Pess, [] P.A. MacMahon, Cetain special patitions of numbes, Q. J. Math. 2 (886),

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