RANKS OF PARTITIONS AND DURFEE SYMBOLS

Transcription

1 The Pennsylvania State University The Graduate School Department of Mathematics RANKS OF PARTITIONS AND DURFEE SYMBOLS A Thesis in Mathematics by William J. Keith c 2007 William J. Keith Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2007

2 ii Committee Page The thesis of William Keith has been reviewed and approved* by the following: Dr. George Andrews Professor, Mathematics Thesis Adviser Chair of Committee Dr. Wen-Ching W. Li Professor, Mathematics Dr. Ae Ja Yee Assistant Professor, Mathematics Dr. Martin Fürer Professor, Computer Science Dr. John Roe Professor, Mathematics Department Head, Mathematics *Signatures are on file in the Graduate School.

3 iii Abstract This thesis presents generalizations of several partition identities related to the rank statistic. One set of these is new: k-marked Durfee symbols, as defined in a paper by Andrews. This thesis extends and elaborates upon several congruence theorems presented in the paper that originated those objects, showing that an infinite family of such theorems exists. The number of l-marked Durfee symbols of n are related to the distribution of ranks of partitions of n modulo 2l + 1; the relationship is made explicit and explored in various directions. Another set of identities deals with the very classical theorem of Euler on partitions into odd and distinct parts. This was given bijective proof by Sylvester, giving occasion to discover new statistical equalities, which in turn were generalized to partitions into parts all c (mod m) by Pak, Postnikov, Zeng, and others. This work further extends the previous theorems to partitions with residues (mod m) that differ but do not change direction of difference, i.e. residues monotonically rise or fall. Attached as an appendix is a translation of the thesis of Dieter Stockhofe, Bijektive Abbildungen auf der Menge der Partitionen einer naturlichen Zahl. This is provided in support of the tools therefrom used in Chapter 3, as well as in the spirit of a service to the Anglophone mathematical community.

4 iv Table of Contents List of Figures vi Acknowledgments viii Chapter 1. Introduction Durfee Symbols Fine s Theorems Chapter 2. The Full Rank: Congruences and Complete Behavior Prime Moduli Nonprime Moduli Chapter 3. Generalizing Sylvester s Bijection Definitions Sylvester s Map Generating Functions Appearance of Descents Appendix A. Notation Appendix B. Translation: Stockhofe s Thesis B.1 Foreword B.2 Notation

5 v B.3 The q-modular Diagram B.4 Construction of L q B.4.1 q-flat Partitions B.4.2 A Smaller Bijective Transformation B.4.3 A Larger Transformation B.4.4 Generalizing Conjugation B.4.5 The Bijection L q B.5 Some Counting Theorems B.6 The Special Case q = B.7 The Fixed Points of L q B.8 Groups of Permutations of P (n) References

6 vi List of Figures 1.1 The Ferrers diagram The m-modular diagram Illustration of Illustration of η The two-residue case Cases of with descents in ρ B.1 Part (i) B.2 Part (ii) B.3 A column inserted B.4 An angle inserted B B B.7 A sketch of the process B B B B B B

7 B B vii

8 viii Acknowledgments I am foremost indebted to my advisor, Dr. George Andrews, for the mathematical mentoring he has provided and for generous support through a long graduate career.

9 1 Chapter 1 Introduction In this chapter we lay out the basics of the theory of partitions and provide at least the minimal definitions and toolkit necessary for any reader to understand what this thesis is about and how it relates to previous work. We say an integer vector λ = (λ 1,... λ k ) is a partition of n if λ 1 λ k 1 and λ λ k = n. The number of partitions of n shall be denoted p(n). A common method for illustrating a partition, especially when we want to construct bijections between sets of partitions of various types, is the Ferrers diagram, consisting of a lattice of dots, each column representing the i-th part and being of height λ i : Fig The Ferrers diagram and Durfee square of λ = (8, 8, 7, 5, 5, 5, 5, 4, 3, 2, 2, 1). An example of a bijection on partitions immediately suggested by the Ferrers diagram is conjugation, in which we map a partition λ to the partition φ illustrated by

10 2 the diagram of λ transposed about its main diagonal. This common and fundamental procedure we will label throughout this thesis φ = λ. It is clear that conjugation preserves the length of the main diagonal and the size of the largest square that can fit in a partition s diagram from the upper left corner, indicated on the illustration above. This square is called the Durfee square of the partition. Non-diagrammatically, we can state that the size c of the Durfee square in λ is the largest c such that λ c c 0. In the above illustration, c = 5. In his 1944 paper Some Guesses in the Theory of Partitions, Freeman Dyson introduced in 5 pages flat a tool of enormous utility to the field: his rank statistic for partitions, defined quite simply as the largest part of a partition, minus the number of parts. Letting N(m, n) be the number of partitions of n with rank m, the generating function of this statistic is R 1 (z; q) = N(m, n)z m q n = q n2 (1.1) (zq; q) m= n 0 n 0 n (q/z; q) n where (a; q) n = n 1 i=1 (1 aqi ). Common methods in combinatorial theory interpret the right hand side of the equation above as a sum over Ferrers diagrams with Durfee squares of size n (a square starts with rank 0), and with rows of length no more than n below the Durfee square (contributing z each) and columns of length no more than n to the right of the Durfee square (contributing z 1 ). It is easily observed that conjugation negates rank, and so N(m, n) = N( m, n), a fact of which we make use in Chapter 2. Originally, Dyson s introduction of this construction was motivated by the fact that partitions with rank i (mod 5, 7) were distributed evenly for partitions of 5n + 4

11 3 and 7n + 5 respectively: that is, if N(i, p, n) denotes the number of partitions of n with rank i (mod p), then N(i, 5, 5n+4) = N(j, 5, 5n+4) and N(i, 7, 7n+5) = N(j, 7, 7n+5) for all i, j. This provided a combinatorial explanation of Ramanujan s famous theorems that p(5n+4) 0 (mod 5) and p(7n+5) 0 (mod 7). (Though it failed to prove the third theorem, that p(11n + 6) 0 (mod 11); this theorem, and the many related congruences later produced first by Ramanujan and collaborators, and in more recent years by Ken Ono and his school, awaited a related statistic called the crank, which Dyson conjectured but was unable to find. This was done by Frank Garvan, in partial concert with George Andrews, decades later.) Since then the rank has developed in rich and unexpected directions, two of which are studied in this thesis. 1.1 Durfee Symbols In studying further partition congruences, A.O.L. Atkin and Frank Garvan [2] related the rank and the crank via a differential equation, in doing so constructing the k-th moments of the rank function. George Andrews [1] has in turn constructed the symmetrized k-th moment ( m + k 1 ) η k (n) = 2 N(m, n) k m= and associated to these objects the k-marked Durfee symbol, in which a partition is decomposed as described in Equation 1.1 and the columns and rows about the Durfee square are marked with k subscripts or colors, according to the following rules:

12 t Definition 1. The ordered, subscripted vector pair 1 t 2... t r b 1 b 2... b s c Durfee symbol of n = c 2 + t t r + b b s if 4 is a k-marked t i, b j {1 1, 1 2,..., 1 k, 2 1, 2 2,..., 2 k,..., c 1,..., c k }; i > j, t i (resp. b i ) = a b, t j (resp. b j ) = d e a d, b e; Every subscript 1,..., k 1 appears at least once in the top row; If M 1, M 2,... M k 2, M k 1 are the largest parts with their respective subscripts in the top row, then b i = d e d [M e 1, M e ], setting M 1 = 1 and M k = c. If we then call D (n) the number of k-marked Durfee symbols of n, then k D (n) = η (n) (Corollary 13 in [1]). The study of congruence theorems for Durfee k+1 2k symbols thus informs the study of congruence theorems for partitions of standard type. Making this information explicit requires defining a richer rank these objects bear called the full rank, preserving some of the properties of the k-coloration: Definition 2. Let δ be a k-marked Durfee symbol and let τ i (resp. β i ) be the number of parts in the top (resp. bottom) row with subscript i. Then the i th -rank of a Durfee symbol is τ β 1 ρ (δ) = i i i τ β i i 1 i < k i = k. Definition 3. The full rank of a k-marked Durfee symbol δ is ρ 1 (δ) + 2ρ 2 (δ) + 3ρ 3 (δ) + + kρ k (δ).

13 5 We set D (m,..., m ; n) to be the number of k-marked Durfee symbols with i th k 1 k ranks all m. In analogy to our previous construction for the rank we call NF (m, n) the i l number of l-marked Durfee symbols of n with full rank m, and NF (b, p, n) the number l of l-marked Durfee symbols of n with full rank b (mod p). Andrews produces the generating function (Theorems 10 and 7 in [1]): n D (n,..., n ; n)x 1 n... x k q n = R (x,... x ; n) k 1 k 1 k k 1 k n 1,...,n k = n 0 k = i=1 R (x ; q) 1 i. (1.2) k (x x )(1 x 1 xj 1 ) i j i j=1 j i This theorem in hand, he produces two congruences: that D (n) 0 (mod 5) for 2 n 1, 4 (mod 5) and D (n) 0 (mod 7) for n 0, 1, 5 (mod 7), because NF (i, 5, n) = 3 2 NF (j, 5, n) and NF (i, 7, n) = NF (j, 7, n) for all i, j in those progressions. Furthermore, it transpires that for n 1, 4 (mod 5) or n 0, 1, 5 (mod 7), we nevertheless have NF (i, 5, n) = NF (j, 5, n) and NF (i, 7, n) = NF (j, 7, n) for all i, j 0 in any progression. l l l l It is our intent in the next chapter to put the above two theorems in a more general setting. We show that they are the simplest two examples of an infinite family of related theorems; we explore the failure mode of the latter cases and explain by exactly how much they fail, giving rise to an infinite family of congruences for prime modulus;

14 and we examine to full detail the behavior of the residue classes for nonprime (odd) modulus Fine s Theorems Chapter 3 of this thesis establishes identities that at once refine and generalize the rank, combining lines pursued separately by previous authors, particularly Glaisher and Fine. In the inaugural work of partition theory, Chapter 16 of Introductio in Analysin Infinitorum, Leonhard Euler shows that the number of partitions of n into odd parts are equinumerous with those in which parts are distinct. Glaisher later generalized this to partitions into parts not divisible by m; N.J. Fine refines it by showing that the number of partitions of n into odd parts, with largest part plus twice the number of parts equal to 2M + 1, equals the number of partitions of n into distinct parts with largest part M. The penultimate statistic here can be regarded as one instance of a generalization of the rank, the (a, b)-rank: a times the largest part of a partition, minus b times the number of parts. Here, we have the (1, 2)-rank. Fine s proof is via generating functions; the theorem can also be proven bijectively by a transformation of Sylvester, an m-modular generalization of which becomes our primary tool to prove a similar theorem for the (1, m)-rank. In exploring this a different diagrammatic presentation of partitions is useful: the m-modular diagram. In this presentation, we fix a modulus m and display each λ i = k i m + j i by writing a column consisting of k i repetitions of m, topped (or founded,

15 in which case we present the diagram marked by *) with j i, if j i is nonzero (0 j i < m). Figure 1.2 provides an example. 7 Fig The 5-modular diagram of λ = (22, 19, 15, 13, 7, 6, 2). It will be quickly observed that conjugation is no longer a simply-defined operation. An m-modular analogue of conjugation was produced in 1981 as the doctoral thesis of one Dieter Stockhofe [13], in the process producing several useful tools we employ in Chapter 3. Also, the notion of a Durfee square is necessarily somewhat coarser for an m-modular diagram; denoted d m (λ) (resp. d m (λ)), we can define its size as the largest i such that λ i m i (resp. λ i m i). On the other hand, when m > 1 (m = 1 gives us the original Ferrers diagram) there are more interesting statistics regarding the parts than simply their number. We can treat the list of nonzero residues (mod m) of such parts appearing as a multiset in [1,..., m 1] and construct a more structured statistic taking into account combinatorial statistics on: the number of nonzero residues, the number of kinds of residues appearing,

16 8 the number of descents in the list of residues read from left to right, and perhaps most interestingly sequences of consecutive parts. The observer will note that when m = 2, as in the case for odd partitions, the residues are all 1, and thus the number of kinds of parts is always 1, and the number of descents is necessarily 0; this behavior hides a degeneracy of these statistics which flowers to great effect in the case for higher modulus. In Chapter 3 we examine the work of Sylvester, Pak, Postnikov, Zeng, and others on (m, c) partitions, which are partitions into parts not divisible by m in which all parts have the same residue c (mod m). Their work is a step toward Glaisher s generalization of Euler s theorem, but has the same degeneracies in the number of kinds of parts and the number of descents as those discussed for the m = 2 case. We do not yet obtain the full Glaisher-style generalization for the statistic involving sequences of parts, but we can describe a more general theorem for m-falling or m-rising partitions, in which residues of parts 0(mod m) ascend or descend monotonically.

17 9 Chapter 2 The Full Rank: Congruences and Complete Behavior 2.1 Prime Moduli In the previous chapter, we observed two previously-proven theorems on the congruence behavior of the full rank of 2-marked and 3-marked Durfee symbols, in arithmetic progressions mod 5 and 7 respectively. These are specific instances of a general theorem for the full ranks of k-marked Durfee symbols in arithmetic progressions of any odd modulus: Theorem 1. Let p = 2l + 1 Z, p 5. Say NF l (j, p, pn + d) is the number of l-marked Durfee symbols of pn + d with full rank congruent to j mod p. Then, if gcd(i, p) = gcd(j, p), we have NF l (i, p, pn + d) = NF l (j, p, pn + d). The inverse does not hold. More generally, when p is not prime the investigation of the differences between divisor-groups of residue classes is itself interesting. As a corollary of this theorem, since all residues not congruent to zero are coprime to a prime modulus we have the near-equidistribution Corollary 1. If p = 2l + 1 is prime, p 5, then NF l (i, p, pn + d) = NF l (j, p, pn + d) for all i, j 0 mod p.

18 10 This is the case for the two theorems previously discussed. The additional behavior of complete equidistribution in residue classes comes about due to a second consequence that will be easily seen from the theorem s method of proof: Theorem 2. If p = 2l + 1 is prime, p 5, then NF l (0, p, pn + d) NF l (1, p, pn + d) = N(l 1, p, pn + d) N(l, p, pn + d). Combining Corollary 1 and Theorem 2, then, Corollary 2. If p = 2l + 1 is prime, p 5, then NF (m, n) = D (n) N(l 1, p, n) N(l, p, n) (mod p). l l m= Because this difference is 0 for p = 5, d = 1, 4 and p = 7, d = 0, 1, 5, we have full equidistribution and a clean congruence theorem in those progressions. Proof of Theorem 1. Our basic strategy, as in [1], is to observe p 1 NF n=1 b=0 l (b, p, n)ζ b n q, p where ζ is a primitive p-th root of unity. To prove the general theorem requires the p additional observation that this sum, in terms of the rank, behaves well with respect to sums of conjugate powers of ζ. To make this precise, we break the sum down thus: p

19 11 p 1 b n 2 l NF (b, p, n)ζ q = Rl (ζ, ζ,..., ζp ; q) l p p p n=1 b=0 ( ) i l R ζ ; q 1 p = l i=1 (ζ i ζp j ) ( 1 ζ i j ) j=1 p p j i 1 l l = i j (ζ ) ( i j ) p 1 p 1 ζp 1 ζ ik pn+d p p ζ N(k, p, pn + d)q p i=1 j=1 n=1 k=0 d=0 j i 1 l l = i j (ζ ) ( i j ) ζp 1 ζ p p i=1 j=1 j i p 1 q d p 1 ik pn ζ N(k, p, pn + d)q. p d=0 n 1 k=0 (2.1) Following Atkin, we define r a,b (q; p; d) = n 0 qn (N(a, p, n) N(b, p, n)). Then, for any given d, N(l, p, pn + d)q pn = n 1 N(l 1, p, pn + d)qpn r (q p ; p; d) l 1,l n 1 = n 1 N(l 2, p, pn + d)qpn r (q p ; p; d) l 2,l =... = n 1 N(0, p, pn + d)qpn r 0,l (q p ; p; d).

20 We further note that the evenness of the rank generating function for partitions (a classic example of a bijective proof: conjugate the partition) gives us the identities N(l, p, pn+d) = N(p l, p, pn+d) and thus r b,c (q p ; p; d) = r p c,p b (q p ; p; d). Combined with the previous line and the fact that p 1 ζ b = 0, we have b=0 p 12 N(0, p, pn + d)q pn r (q p ; p; d) i 0 ζ 0,l p n 1 + N(1, p, pn + d)q pn r (q p ; p; d) i 1 ζ ,l p n 1 + N(l, p, pn + d)q pn i l ζ + N(l + 1, p, pn + d)q pn i (l+1) ζ p p n 1 n N(p 1, p, pn + d)q pn r (q p ; p; d) i (p 1) ζ = 0. (2.2) 1,l p n 1 (For use in a later theorem we note that it matters in the above calculation that i 0 mod p in this context, but its value otherwise is irrelevant; if p is nonprime and gcd(i, p) 1, we have merely employed the same identity Thus, gathering the N(k, p, pn + d) terms, p gcd(i,p) times.) p 1 ζ pik N(k, p, pn + d)q pn = r (q p l 1 ; p; d) + r (q p ( ig i( g) ) ; p; d) ζ + ζp. 0,l g,l p k=0 n 1 g=1

21 13 Thence 1 p 1 b n l l NF (b, p, n)ζ q = i j l p (ζ ) ( i j ) ζp 1 ζ p p n=1 b=0 i=1 j=1 j i p 1 q d r (q p l 1 ; p; d) + r (q p ig i( g) ; p; d) (ζ ) + ζp. (2.3) 0,l g,l p d=0 g=1 For any n, then, we have by equation of coefficients in powers of q that 1 p 1 b l l NF (b, p, n)ζ = i j l p (ζ ) ( i j ) ζp 1 ζ p p b=0 i=1 j=1 j i l 1 ig i( g) N(0, p, n) N(l, p, n) + (N(g, p, n) N(l, p, n)) (ζ ) + ζp. (2.4) p g=1 To prove the theorem, it suffices to show that the right-hand side of 2.4 is an integer. The constant term that appears before the sum contributes only 0: notice that l i j (ζ ) ( i j ) j=1 ζp 1 ζ i(l 1) = ζ l i+j (1 ) ( i j ) ζ 1 ζ, p p p j=1 p p j i j i and the exponents { i + j, i j 1 j l, j i} are precisely {1,..., p 1} \ {0, i, 2i} when reduced mod p. Since ( p 1 i ) 1 ζ = p, we can simplify the term thus: i=1 p

22 14 1 l l (N(0, p, n) N(l, p, n)) i j (ζ ) ( i j ) ζp 1 ζ p p i=1 j=1 j i = (N(0, p, n) N(l, p, n)) 1 l p i(l 1) ( 2i ) ( i ) ζ 1 ζ 1 ζ p p p i=1 = (N(0, p, n) N(l, p, n)) 1 l p i(l 1) i(l+2) i(l+1) i(l) (ζ ) + ζp ζp ζp p i=1 = (N(0, p, n) N(l, p, n)) 1 l p i(l 1) i(l 1) i(l+1) i(l+1) (ζ ) + ζp ζp ζp p i=1 = (N(0, p, n) N(l, p, n)) 1 2l p i(l 1) i(l+1) (ζ ) ζp p i=1 = (N(0, p, n) N(l, p, n)) 1 ( 1 ( 1)) = 0. (2.5) p

23 15 There remains the second term, which contributes a nonzero integer: 1 l l i j (ζ ) ( i j ) l 1 ζp 1 ζ ig i( g) p p (N(g, p, n) N(l, p, n)) (ζ ) + ζp p i=1 j=1 g=1 j i l i(l 1) ( 2i ) ( i ) 1 l 1 ig i( g) = ζ 1 ζ 1 ζ (N(g, p, n) N(l, p, n)) (ζ ) + ζp p p p p p i=1 g=1 = 1 l l 1 p [ i(l g 1) i(l+g 1) i(l g+2) (N(g, p, n) N(l, p, n)) ζ + ζp + ζp p i=1 g=1 +ζ p i(l+g+2) ζp i(l g) ζp i(l+g) ζp i(l g+1) ζp i(l+g+1) ] = 1 l 1 p l [ i(l g 1) i(l g+2) i(l g+2) (N(g, p, n) N(l, p, n)) ζ + ζp + ζp p g=1 i=1 +ζ p i(l g 1) ζp i(l g) ζp i(l g+1) ζp i(l g+1) ζp i(l g) ] = p 1 l 1 p 1 i(l g+2) i(l g 1) i(l g+1) i(l g) (N(g, p, n) N(l, p, n)) [ζ ] + ζp ζp ζp p g=1 i=1 = 1 l 1 p (N(g, p, n) N(l, p, n)) ɛ, (2.6) g=1 where ɛ = 0 if g l 1 and ɛ = p if g = l 1. Thus, the right-hand side of 2.4 is an integer, and so 2.4 is a polynomial of degree p 1 in ζ p over the integers. We can particularly evaluate p 1 b=0 NF l (b, p, n)ζ p b = N(l 1, p, n) N(l, p, n). (2.7) From the properties of primitive roots, the theorems and corollaries follow.

24 16 The behavior of ɛ explains theorem 2. Work of Atkin and Swinnerton-Dyer [4] yields the arithmetic progressions mentioned by Andrews, for p = 5 and p = 7, in which the difference N(l 1, p, pn + d) N(l, p, pn + d) is identically 0 and equidistribution of the l-ranks is achieved. A study of the difference N(l 1, p, n) N(l, p, n) has been made for additional prime p by Atkin and collaborators Hussain [3] and O Brien [8]: specifically p = 11, 13, 17, and Nonprime Moduli We now turn to a deeper examination of nonprime p. No longer is the polynomial 1 + x + x x p 1 irreducible over the integers, so the populations of the various divisor-groups of residue classes mod p are no longer necessarily equal. However, if we can establish N(0, p, n) N(d, p, n) for all d p, we can state a congruence theorem for D (n) modulo p. l d 2d ld We do this by observing the behavior of R (ζ, ζp,..., ζp ). From Theorem l p 9 of [1], we have d 2d ld R (ζ, ζp,..., ζp ) = q n l p NF (j, p, n) p n 0 l j j p p dj + j p p dj NF l (j, p, n)µ (gcd(p, dj)) gcd(p, dj) j k prime k p j (1 1 k ) k prime,k p/j k p/gcd(p,dj) (1 1 k ) (2.8)

25 where µ is the standard Möbius function. (The expression appears involved, but calcula tion for any given p is not difficult. By way of example we use later, R (ζ, ζ, ζ9, ζ9 ; q) = ( ) n 0 qn NF (0, 9, n) + 0 NF (1, 9, n) NF (3, 9, n), and R (ζ, ζ9, ζ9, ζ9 2; q) = ( ) n 0 qn NF (0, 9, n) 3NF (1, 9, n) + 2NF (3, 9, n).) Calculating this value for each d strictly dividing p gives us a system of d(p) 1 linear equations in the N(d, p, n) (where d(p) is the divisor function) that we can solve explicitly for the differences N(0, p, n) N(d, p, n). The primary obstacle to this calculation is that we cannot simply assign x i = ζ p di in Theorem 7 of [1], as we did with d = 1 in the theorem above. Doing so produces singularities in the terms 1 (x i x j )(1 x 1 i x 1 when j ±i (mod p/d). These singularities j ) are, of course, removable; the problem of evaluation is simply to do so, and the method is repeated application of L Hopital s rule. The case p = 9 is the first opportunity to employ the method, the most tractable to calculate explicitly for illustrative purposes, and an interesting example in its own right. We begin with Theorem 7 itself: 17 k R (x, x, x, x ; q) = i=1 k j=1 j i R 1 (x i ; q) (x i x j )(1 x 1 i. x 1 ) j We know that

26 18 R 4 (ζ 9, ζ 9 2, ζ9 3, ζ9 4 ; q) = n 0 q n ( ) NF (0, 9, n) NF (3, 9, n) 4 4 = n 0 q n (N(3, 9, n) N(4, 9, n)). (2.9) Already we can state an interesting congruence: a conjecture of Richard Lewis [7] proved by Nicholas Santa Gadea [11] states that N(3, 9, 3n) = N(4, 9, 3n). Thus NF (0, 9, 3n) = NF (3, 9, 3n) = NF (6, 9, 3n) and, since NF (i, 9, n) = NF (j, 9, n) for the 6 residue classes 3 i, j, we have Theorem 3. D 4 (3n) 0 (mod 3). To say more regarding the behavior of D 4 (mod 9), we need to know the difference NF 4 (0, 9, n) NF 4 (1, 9, n). To obtain this we wish to calculate, for d = 3, R 4 (ζ 9 3, ζ9 6, ζ9 9, ζ9 3 ; q) = R4 (ζ 3, ζ 3 2, 1, ζ3 ; q) = q n ( ) NF (0, 9, n) 3NF (1, 9, n) + 2NF (3, 9, n) n 0 (2.10) in terms of R (ζ ; q). 1 3 Our strategy is to replace, one by one, each of the x by functions of x which i 1 i replicate the relations of the ζ : x4 by x, x by 1, and x by x 1. At each step we obtain a small number of singularities we can remove. First, let us replace x by x. 4 1

27 19 R 4 (x 1, x 2, x 3, x 1 ; q) = = lim x 4 x 1 + lim x 4 x 1 R 4 (x 1, x 2, x 3, x 4 ; q) R (x ; q) 1 1 (x x )(x x )(x x )(1 x x 1 )(1 x x 1 )(1 x x 1) 4 R (x ; q) 1 4 (x x )(x x )(x x )(1 x x 1 )(1 x x 1 )(1 x x 1) 3 = + + R 1 (x 2 ; q) (x 2 x 1 ) 2 (x 2 x 3 )(1 x 1 2 x 1 1 )2 (1 x 1 2 x 1 3 ) R 1 (x 3 ; q) (x 3 x 1 ) 2 (x 3 x 2 )(1 x 1 3 x 1 1 )2 (1 x 1 3 x 1 2 ) R 1 (x 2 ; q) (x 2 x 1 ) 2 (x 2 x 3 )(1 x 1 2 x 1 1 )2 (1 x 1 2 x 1 3 ) + lim x 4 x 1 R (x ; q) (x x ) 2 (x x )(1 x x 1 1 )2 (1 x 1 3 x 1) 2 1 (x x )(1 x x 1) 1 i=1,4 j=2,3 1 (x x )(1 x 1 i j i x 1 ) j ( R (x ; q)(x x )(x x )(1 x x 1 )(1 x x 1) 3 )) R (x ; q)(x x )(x x )(1 x x 1 )(1 x x 1) 3. (2.11) After differentiation and taking the limit, we obtain

28 20 R 4 (x 1, x 2, x 3, x 1 ; q) = + R 1 (x 2 ; q) (x 2 x 1 ) 2 (x 2 x 3 )(1 x 1 2 x 1 1 )2 (1 x 1 2 x 1 3 ) R 1 (x 3 ; q) (x 3 x 1 ) 2 (x 3 x 2 )(1 x 1 3 x 1 1 )2 (1 x 1 3 x 1 2 ) x R (x ; q) (x x )(x x )(1 x x 1 )(1 x x 1 )(1 x 2 ) 3 1 R (x ; q)( x 1 x + 2 x 1 1 x + x 2 1 x x 1 + x 2 1 x x x 1 ) 1 x 1 3 (x x )(x x )(1 x x 1 )(1 x x 1 )(1 x 2 ). (2.12) 3 1 For the next step we replace x 3 by 1. In the case of d = 3, replacing x p/3 by 1 produces no singularities, and so we need not differentiate. (This is the only divisor where this degeneracy ever occurs; for any other potential divisor of p, d l > 2 means that this replacement step would produce singularities in the denominator factors (x kp d and (1 x 1 x 1 ).) For p = 9, we obtain kp hp d d x hp ) d

29 21 R 4 (x 1, x 2, 1, x 1 ; q) = R 1 (x 2 ; q) (x 2 x 1 ) 2 (x 2 1)(1 x 1 2 x 1 1 )2 (1 x 1 2 ) + R 1 (1; q) (1 x 1 ) 2 (1 x 2 )(1 x 1 1 )2 (1 x 1 2 ) x R (x ; q) (x x )(x 1)(1 x x 1 )(1 x 1)(1 x 2 ) R (x ; q)( x 1 x + 2 x x 2 1 x x 1 + x x x 1 ) 1 (x x )(x 1)(1 x x 1 )(1 x 1)(1 x 2 ). (2.13) It remains to replace x 2 by x R (x, x 1, 1, x ; q) = lim x x 1 R(x, x, 1, x ; q) = lim x 2 x 1 1 (x 1 x 2 ) 2 (x 1 1)(1 x 1 1 x 1 1 R (x ; q) R (x ; q)( + 1 x x x x R 1 (1; q) (1 x 1 ) 3 (1 x 1 1 )3 2 )2 (1 x 1 )(1 x 1)(1 x 2 )(x ) R (x ; q)((x 1)(1 x 1 )(1 x 2 )) + ((x x )(1 x x 1)(x x 1 x x 1 1 x )(1 x 1 2 )) x x 1 1 ). (2.14)

30 22 We differentiate (twice) with respect to x 2 and note the identity lim x 2 x x 2 R 1 (x 2 ; q) = x x 2 R 1 (x 1 ; q) + 2x 3 R (x ; q) 1 x to obtain in the limit R 4 (x 1, x 1 1, 1, x 1 ; q) = R 1 (1; q) (1 x 1 ) 3 (1 x 1 1 )3 + x 1 4 2(1 x 1 ) 3 (1 x 1 1 ) 3 (1 x1 2 ) 3 2 x 2 R 1 (x 1 ; q)x 4 (x1 1) (1 x ) (1 x1 ) x 1 R 1 (x 1 ; q)(1 x 1 ) 2 (1 x 1 1 )2 ( x 1 3 2x1 2 2x1 ) ] R (x ; q)(1 x ) (1 x1 ). (2.15) We have now removed all the troublesome singularities and can set in the last identity x 1 = ζ 3 to evaluate

31 23 R 4 (ζ 3, ζ 3 2, 1, ζ3 ; q) = n 0 q n (NF 4 (0, 9, n) 3NF 4 (1, 9, n) + 2NF 4 (3, 9, n)) = R (1; q) ζ (1 ζ 3 ) 3 [ 2 2 ] 9(ζ ζ ) 3 3 z 2 R (z; q) z=ζ3 z R 1 (z; q) z=ζ (1 ζ )R1 (ζ ; q) 3 3 ( = ) 2R (1; q) + 3ζ z 2 R (z; q) + 2(ζ 1) 1 z=ζ3 3 z R 1 (z; q) z=ζ3 2R (ζ ; q). 1 3 (2.16) We wish to rewrite this formula in terms of the rank classes N(j, n). The termwise first and second derivatives of N(j, n)z j q n, jn(j, n)z j 1 q n and j(j 1)N(j, n)z j 2 q n respectively, group themselves thus by the residue class of j modulo 3 when evaluated at z = ζ : 3

32 24 R 4 (ζ 3, ζ 3 2, 1, ζ3 ; q) = 1 q n n ( (27k 2 3k)N(3k, n) 6kN(3k + 1, n) + 2N(3k + 2, n) 54 n 0 k= n = 1 q n 54 n 0 n k= n + ζ ((27k 2 ) + 15k)N(3k + 1, n) (6k + 4)N(3k + 2, n) 3 2 ( + ζ (27k 2 )) + 33k + 8)N(3k + 2, n) 6kN(3k, n) 3 (27k 2 + 3k)N(3k, n) 6kN(3k + 1, n) (27k k + 6)N(3k + 2, n) + ζ (6kN(3k, n) + (27k k)N(3k + 1, n) (27k 2 ) + 39k + 12)N(3k + 2, n). 3 (2.17) Here we pause to observe that we can simplify the sum above by recalling that, due to the evenness of the rank function, for any j j j kn(3k, n) = kn(3k + 1, n) + (k + 1)N(3k + 2, n) = 0 k= j k= j and j k 2 N(3k + 1, n) = j (k + 1) 2 N(3k + 2, n). k= j k= j With these two identities the ζ 3 term of 2.17 wholly vanishes. (We knew it must, since of course R 4 (ζ 3, ζ 3 2, 1, ζ3 ; q) has integral coefficients.)

33 Upon discarding the vanishing ζ 3 term and simplifying the remainder with the relations above we have that 25 NF 4 (0, 9, n) 3NF 4 (1, 9, n) + 2NF 4 (3, 9, n) (( = 1 n 9k 2 ) ( N(3k, n) 9 2 k= n ( n k 2 = 2 = k= n n k 2 N(3k, n) k=1 (9k 2 ) ) + 9k) N(3k + 2, n) 2 ) k(k + 1) N(3k, n) N(3k + 2, n) 2 k(k + 1) 2 (N(3k + 1, n) + N(3k + 2, n)). (2.18) Thus, since we already know NF 4 (0, 9, n) NF 4 (3, 9, n) = N(3, 9, n) N(4, 9, n), NF (0, 9, n) NF (1, 9, n) = 2 (N(3, 9, n) N(4, 9, n)) n k 2 N(3k, n) 3 k=1 k(k + 1) 2 (N(3k + 1, n) + N(3k + 2, n)). (2.19) Putting these all together, we have

34 26 D 4 (n) = NF 4 (0, 9, n) + 6(NF 4 (0, 9, n) (NF 4 (0, 9, n) NF 4 (1, 9, n))) + 2(NF 4 (0, 9, n) (NF 4 (0, 9, n) NF 4 (3, 9, n))) = 9NF 4 (0, 9, n) + 2(N(3, 9, n) N(4, 9, n)) n 2k 2 N(3k, n) k(k + 1)(N(3k + 1, n) + N(3k + 2, n)) k=1 2(N(3, 9, n) N(4, 9, n)) n 2k 2 N(3k, n) k(k + 1)(N(3k + 1, n) + N(3k + 2, n)) (mod 9). (2.20) k=1 For n 0, 1, 2 (mod 3), the identities of [11] provide specializations of this identity when we dissect the sum over k by the residue classes of k modulo 3. In the case of general p and divisor d, we perform variable replacements patterned on those we saw above. We replace x i+ kp d finally replace x p with x 1 d i i with x i for 0 < i < p d, replace x kp d with 1, and for 0 < i p 2d. We eventually encounter derivatives of order up to 2d, in order to clear singularities. When we then evaluate Theorem 7 at x i = ζ p di, a great deal of simplification can occur by working with the evenness of the rank function. A general form for these functions should be easy to obtain.

35 27 Chapter 3 Generalizing Sylvester s Bijection One of the first theorems the student of partitions learns is Euler s theorem that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. This simple statement has been refined, generalized, and expanded upon by many workers in the field. In 1883, Glaisher showed this to be the m = 2 case of a general theorem equating the number of partitions of n into parts not divisible by m to partitions into parts appearing fewer than m times. J.J. Sylvester s student Franklin made it the k = 0 case of a theorem equating the number of partitions of n with k sizes of even part, and those with k sizes of repeated part. In 1882, Sylvester published his famous paper [14], containing his bijective proof of Euler s theorem. Later authors starting with Cayley and more recently including Bessenrodt [5], Fine [6], Kim and Yee [15], and Pak and Postnikov [10], to name a few have found refinements of Euler s theorem via this bijection. The transformation not only maps partitions with odd parts to those with distinct parts, but also preserves a number of statistics on partitions of each type, such as the number of parts in the starting odd partition λ and the alternating length of the target 2-distinct partition µ (defined as µ µ +µ µ +..., and easily seen to be number of odd parts in the conjugate µ) The introduction of these statistics allows the construction of finer identities equating the generating functions of partitions with parameters for each statistic.

36 28 The challenge this chapter sets is to combine these two lines of development, and carry the refined statistics present in Sylvester s bijection from partitions into odd parts toward partitions into parts not divisible by m. We begin with an identity due to N.J. Fine. In [6], he proves that: Theorem 4. (Fine) Partitions of n into distinct parts with largest part M are equinumerous with partitions of n into odd parts with largest part plus twice the number of parts 2M + 1. Theorem 4 is numbered in [6]. The following corollary actually appears earlier, as equations 23.8, but Fine points out that it can be deduced from by noting that the number of parts in a partition of n into odd parts has the same parity as n: Corollary 3. The number of partitions of n into distinct parts with largest part a (mod 2), a = 0, 1, is equal to the number of partitions of n into odd parts with largest part 2a + 1 (mod 4) if n is even, and 2a 1 (mod 4) if n is odd. Fine proved his theorem analytically, but Pak and Postnikov in [10] show that this, and additional statistics, can be proved with a generalization of Sylvester s bijection. (They also generalize these statistics to the (m, c) case, as we will discuss momentarily.) The most general collection of statistics appears to be in a paper of Zeng [16] published in Denote by l(λ) be the number of parts in λ a partition into odd parts and by d (λ) the largest i such that (λ 1)/2 i. Then from that paper, we have: 2 i Theorem 5. Let λ be a partition of n into odd parts and φ be Sylvester s bijection, µ = φ(λ). Then µ is a partition of n into distinct parts; l(λ) equals the alternating

37 length of µ; l(λ) + (λ 1 1)/2 = µ 1 ; the number of distinct parts in λ equals the number of sequences of consecutive parts in µ; and d 2 (λ) = l(µ)/2. 29 The one remotely successful attempt at generalization of Sylvester s bijection toward Glaisher s theorem appears to be the (m, c)-analogues of the bijection. These treat what are called (m, c) partitions those in which parts are all congruent to c (mod m). These are equinumerous with partitions of type (c, m c, c, m c,...), where a partition is of type (a, a,... ) if the largest part appears a times, the next largest part appears a times, etc. There are also sets of related statistics: the same as in the previous theorem, except that in general we replace 2 with m, such as using d (λ) (replacing (λ 1)/2 m i with (λ c)/m) and alternating length µ µ + µ µ (Readers of i 1 m m+1 2m Zeng s paper should be careful about the latter, which is not clarified.) However, this certainly falls short of treating all partitions into parts 0 (mod m). More importantly, Zeng points out that the identity thus obtained (Theorem 4 in [16]) is algebraically equivalent to the original identity with a simple substitution of variables. The reason for this is that, with regard to the characteristics manipulated by the bijection, any (m, c) partition has the same m-modular shape regardless of what the m and c actually are. An algebraically richer identity thus requires considering partitions into parts not divisible by m in which different residues of parts modulo m appear. In this chapter are established such identities in the cases where residues mod m increase weakly monotonically from the smallest part to the largest, and those in which the residues decrease monotonically. To be specific, we establish, where f is the λ number of descents among the nonzero residues (mod m) of parts of λ read right to left as a multiset on [1,..., m 1]:

38 30 Theorem 6. The number of partitions µ of n into parts appearing fewer than m times, with largest part µ, is equal to the number of partitions λ of n into parts not divisible 1 by m with λ + m (l(λ) f ) = m µ + j, 0 < j < m. 1 λ 1 Summation in residue classes gives us Corollary 4. The number of partitions µ of n into parts appearing fewer than m times, with largest part µ 1 b (mod m), is equal to the number of partitions λ of n into parts not divisible by m with λ 1 + m (l(λ) f λ ) mb + j (mod m 2 ), 0 < j < m. We actually get more (for the full statement, see Theorem 9), though the definition and counting of chains of consecutive-or-equal parts requires some delicacy. The (m, c) generalization of Sylvester s bijection can stand up to most of the task, but runs into difficulties when calculating the number of these sequences. We therefore use a different generalization; the next section lays out the definition and the necessary tools for its analysis. 3.1 Definitions In his 1981 thesis [13], Dieter Stockhofe constructed for each n a collection of bijections L which together generate all bijections on the set of partitions of n. Each m,n L is itself a collection of bijections between certain classes of partitions of n with m,n specific characteristics related to their m-modular diagrams. To describe the L more m,n precisely we need the following definitions, mostly from [13]. Fixing α a partition of n, α = (α,..., α ), and a modulus m: 1 k Definition 4. The m-weight of α i is α i m = α i m.

39 We will illustrate the definitions of this section with a continuing example partition α = (42, 39, 30, 25, 23, 20, 16, 10, 7, 5, 5), using the modulus m = 5 unless otherwise noted. 31 Example 1. α 1 5 = 8; α 11 5 = 1. Definition 5. Let β i α i (mod m) be the least nonnegative residue of α i. Then the residue-vector ρ(α) is the r-tuple (ρ 1,..., ρ r ) = (β i1,..., β ir ) of nonzero β i with i 1 < i 2 < < i r. Intuitively, the process to construct ρ consists of removing all parts of α divisible by m and reducing the remaining parts mod m. Example 2. ρ(α) = (2, 4, 3, 1, 2). Definition 6. A part α i of α contains j m-edge units (or simply edge units when m is understood) if (m + 1)j > α i α i+1 mj, setting α k+1 = 0. Intuitively, j measures the amount by which α i exceeds the minimum multiple of m necessary for α to be a partition. Example 3. α has 5-edge units in parts 2, 3, 7, and 11. Observe that the last part contains a 5-edge unit since it is at least 5. Also note that it is possible for a part to have more than one m-edge unit; for example, if we had set m = 2, then α would contain edge units, since α α = In an associated vein, we may speak of an m-strip: a set consisting of the m-edge unit in α and one multiple of m in each larger part. These m-strips can be subtracted i from the parts in α in which they appear, leaving a sequence which is still a partition (m-edge units are defined by being in excess of the minimum allowable size of a part)

40 32 and replaced in the new partition as, for example, a part equal to m times the number of elements in the strip so defined. Reversible examples of such manipulations are used by Stockhofe in defining L. m Example 4. The 5-strips of α are of lengths 2, 3, 7, and 11; there is one of each length. If the set of all m-strips is subtracted from α and collected as a new partition into parts divisible by m, this new partition is called mα, the strips of α. (Addition s and scalar multiplication of partitions are defined componentwise as the standard vector operations for vectors of length equal to the longer partition, filling out the smaller with zeroes; subtraction is defined when the result is still a partition. Once m is fixed we often speak of α and mα interchangeably as suits the context.) s s Example 5. α s = (20, 20, 15, 10, 10, 10, 10, 5, 5, 5, 5). It is perhaps more illuminating to observe that α s = (55, 35, 15, 10) = 5(11, 7, 3, 2). Definition 7. P ρ k,l is the set of partitions that have residue-vector ρ, k parts divisible by m, and a total of l m-edge units (or, equivalently, m-strips). P ρ (n) is the set of such k,l partitions of n. With these definitions, L exchanges P ρ ρ (n) with P (n). It thereby estab- m,n k,l l,k lishes a generalization of conjugation (which is, in fact, L 1,n ) for m-modular diagrams. Later we will consider the actual bijection; for many counting theorems, it is sufficient to recall the somewhat astounding fact that these two classes are equinumerous. In order to use this generalization of conjugation to extend classical identities, we now make a simple additional observation: namely, that when α i = β i + t i m contains j m-edge units, if β i β i+1 then j = t i t i+1, whereas if β i < β i+1, j = t i t i+1

41 1. Viewing ρ as a permutation of a multiset in [1,..., m 1], the number of times ρ i < ρ i+1 the number of descents in the word ρ, to use a more common phrasing in combinatorics is a statistic that relates further useful information about the partition. It is also interconnected with other statistics: for example, each descent increases by 1 the minimum possible m-weight of the largest part of α, and decreases (in a suitably defined average sense) by m 1 m 2 The observer will notice that m 1 m 2 the number of parts in ρ. 33 is undefined for m = 2 and, indeed, makes little sense for m = 1. Naturally, in these cases there are no descents in ρ: for m = 1, ρ is empty, and for m = 2, ρ = 1 r. It is this simplicity that hides the more elaborate combinatorial structure of the generalizations we deal with here. we produce. Finally, we introduce some terminology to smooth the description of the theorems Definition 8. Call ρ,l P ρ 0,l (the set of partitions into parts not divisible by m) m-odd partitions, and ρ,k P ρ k,0 (the set of partitions in which there are no m-strips, or, that have first differences less than m and smallest part less than m) m-flat partitions. Applying traditional conjugation to m-flat partitions we see that these are equinumerous with the set of partitions in which parts appear fewer than m times, and as this gives Glaisher s generalization of Euler s odd-distinct theorem it seems useful to denote these as m-distinct partitions. (The reader should be alerted that the terminology flat is used in at least one other place the author knows of, in Sloane s encyclopedia of integer sequences [12];

42 34 there, flat partitions (Sequence A034296) are partitions with first differences less than or equal to 1 without restriction on the size of the smallest part.) The remaining portion of α, called the flat part of α, can be broken down as m ρ mα f where m ρ is the unique partition in P ρ 0,0 and α f is defined by the following procedure: first, remove from the flat part of α any repeated parts of size divisible by m, but not the last such part (say, part i) of any given size if it lies between parts such that the residues β β. Each removed part becomes a part of α. For the remaining i 1 i+1 f parts divisible by m, remove the intervening parts and prevent m-strips from appearing by removing angles : working from the largest part divisible by m, remove the part itself and conjoin it with the m-strip that results from its removal to form a single part of α. The operation is the (unique) reverse of this process. f Example 6. Let m = 5, α = (42, 39, 30, 25, 23, 20, 16, 10, 7, 5, 5). Then ρ = (2, 4, 3, 1, 2), m ρ = (12, 9, 8, 6, 2), α = (20, 20, 15, 10, 10, 10, 10, 5, 5, 5, 5), and α = (25, 25, 15, 5). s f The operation inserts the parts of α into m ρ in this order: first to insert a weight of f 25, 5 is added to parts 12, 9, and 8 of m ρ to make parts 17, 14, and 13; then a new part 10 is added between 13 and 6. Second, to insert the next 25, 5 is added to parts 17 and 14 to make parts 22 and 19; then a new part 15 is added between 19 and 13. Finally, 15 and 5 are added as parts on their own to create m ρ α = (22, 19, 15, 15, 13, 10, 6, 5, 2), f the flat part of α.

43 Fig α decomposed as in Example 6. 35

44 When one of α f or α s is empty that is, when α is an m-odd or m-flat partition respectively then the bijection L exchanges P ρ ρ (n) with P (n) thus: m,n k,l l,k L m,n ((m ρ mα f ) + mα s ) = (m ρ mα s ) + mαf. Our considerations are solely on relations between m-odd and m-distinct partitions, so this description of the simplified behavior of L m,n on such partitions suffices for our purposes. However, the map is well-defined for all partitions of n, and the interested reader is directed to [13], or the translation in Appendix B, for a fuller description and complete detail of exposition Sylvester s Map In [9], Pak notes that in the literature discussing partitions into odd and distinct parts, numerous bijections appear... and are almost universally the same as that given by Sylvester in his original bijective proof of Euler s theorem. When we observe, therefore, that the composition of Stockhofe s bijections L L is precisely Sylvester s bijection, 1,n 2,n it may seem to be gilding the lily. However, when we note that L L serves the 1,n m,n same purpose for the (m, c)-analogues (the same map suffices for all c for each m), something new finally happens: for while L L is precisely the so-called fishhook 1,n m,n bijection for partitions wherein the residues of parts mod m are all the same, it is not the same map for partitions in which residues differ. To prove these claims we will employ the version of Sylvester s bijection labeled η in [9]. The definition I construct below for general residue-vector ρ simplifies to become the definition so named that Pak uses when applied to odd partitions, and the graphical fishhook map that Zeng uses when applied to (m, c) partitions.

45 37 Definition 9. Fix a modulus m. Let λ be a partition of n with parts 0 (mod m), ρ = (ρ 1,..., ρ k ). Then η(λ) = (..., η mt+i,... ), where with 0 i < m, 0 < t d m (λ), η mt i := λ t m t + #{λ r λ r m t} (t 1) + #{ρ a (a = t or λ a m = t 1) and (m i ρ a )}, and for t = d m (λ) + 1, η mt+i := #{ρ a a t and λ a m = t 1 and (m i ρ a )}. Fig The generalized η: η((17, 14, 13, 12, 6)) = (8, 8, 7, 7, 7, 5, 4, 4, 4, 3, 2, 2, 1) Recall that for a partition into parts not divisible by m, d m (λ) is the largest c such that λ c m c. Another way to interpret this is by removing the residue-vector ρ and regarding the m-modular diagram below it, with the m replaced by the dots of a

46 typical Ferrers diagram: d m (λ) is then the size of the Durfee square of that partition. λ On the other hand, d (λ) uses the largest c such that c m m c it simply includes the residue-vector. In the example above, we have d m (λ) = 2 and d m (λ) = 3. The two values differ exactly when λ dm (λ)+1 = d (λ), when there is an entry of the residue-vector at the m m corner of the m-modular Durfee square. To read η(λ) off of its m-modular diagram [λ] m, we draw hooks through the main diagonal and read their lengths. Here we have [(17, 14, 13, 12, 6)] = Now observe the definition of η. Parts appear in groups of m, in this case 5. In the first hook, we begin with t = 1, and construct parts 1 through 5. The 5-weight of λ 1 is 3, from which we subtract 1 so as not to overcount the corner of the hook; there are 5 parts of 5-weight at least 1, from which we subtract t 1 parts so as not to recount parts larger than λ (of course, there are none yet). So far, we have =7. There is one 1 ρ with a = t, i.e., ρ = 2. There are no parts with λ = 0. So there are 2 parts in a 1 a m η(λ) of size 1 greater than 7: thus, the first 5 parts are (8, 8, 7, 7, 7). These correspond to the 5 outermost hooks in the diagram above. One can immediately see from this description why the largest part of η(λ) will have size equal to the m-weight of the largest part of λ, plus the number of parts of λ

47 39 or, if we scale by m, that m times the largest part of η will be roughly λ plus m times 1 the number of parts of λ. The second group of 5 parts is constructed similarly. The 5-weight of λ is 2; we 2 subtract 2, one to not overcount the corner and one so as not to recount the first row; there are 4 parts of 5-weight at least 2, from which we ignore the first part, so these five parts are of size at least =3. There is one residue for λ, which is 4, and one 2 residue of a part with 5-weight exactly 1, namely ρ = 1. Thus, the first 4 parts are of 5 size at least 1 greater than 3, and the first of the five is 2 greater: the first 10 parts of η(λ) are (8, 8, 7, 7, 7, 5, 4, 4, 4, 3). Finally, the last group is different since we are at the outer corner of the 5- modular Durfee square. The last group of 5 (or fewer) parts counts the number of remaining residues on parts with 5-weight 2 of size at least i: our residues are 3 and 2, so we have 3 parts of size at least 1, and another 2 of size at least 2. Thus, η(λ) = (8, 8, 7, 7, 7, 5, 4, 4, 4, 3, 2, 2, 1). We can now prove that L L (λ) = η(λ) for (m, c) partitions. The proof 1,n m,n illuminates several aspects of the behavior of L L that will be useful to recall in 1,n m,n the more general case. Theorem 7. Given a modulus m and a partition λ with ρ = (c, c,..., c), L 1,n L m,n (λ) = η(λ). Proof. If d m (λ) 1, the largest c parts of η(λ) are η 1 = = η c = λ 1 m 1 + #{λ r λ r m 1} 0 + #{ρ a (a = 1 or λ a m = 0)} = λ 1 m + k, and if d m (λ) = 0 then η 1 = = η c = #{ρ a a 1 and λ a m = 0)} = λ 1 m + k (= k).

48 40 If d (λ) = 0, then L (λ) = λ and so φ = L L (λ) = (k,..., k) = (k c ). m m,n 1,n m,n If d (λ) 1, then φ = = φ = l(l (λ) ) = l(λ ) + l(ρ) = λ + k. So the first m 1 c m,n s 1 m c parts match. If d (λ) = 0, then there are no more parts and we are done. m If d (λ) 1, the next m c parts of η(λ) are η = = η = λ 1 + m c+1 m 1 m #{λ λ 1} = l(λ ) 1 + λ. The next m c parts of φ are the index of r r m s s1 the second-largest part in L (λ), which resulted from inserting the largest part of λ m,n s at the beginning of the operation. This insertion produced a part of size m at index λ, adding m to each previous part; the remaining l(λ ) 1 parts of λ were inserted s1 s s at various lower indices. Thus the index of the last m in L (λ), which becomes the m,n value of the parts φ = = φ, is l(λ ) 1 + λ. c+1 m s s1 If d (λ) = 1, the next c parts of η(λ) have t = d (λ) + 1 and so are η = m m m+1 = η = #{ρ a 2 and λ = 1} = λ 1. In this case as well, the next c m+c a a m s1 parts of φ are the index of the largest of the parts m + c which resulted from adding m to the first λ 1 entries, before inserting any number of parts m to the right of s1 those entries. (It is possible, of course, that λ = 1, i.e. λ = l(λ )m + c, in which case s1 s all of these entries are 0.) Thus η = φ and, as these are the last entries, we 2m i 2m i are done. The next case should be sufficient to illuminate the pattern. If d (λ) 2, m η = = η = λ 2 + #{λ λ 2} 1 + #{ρ (a = 2 or λ = 1)} m+1 m+c 2 m r r m a a m = λ 2 + #{λ λ 1}. On the other hand, φ = = φ are the 2 m r r m m+1 m+c index of the third-largest part of L (λ), which is the part m + c resulting from the m,n addition of m to each part c of m ρ of index less than λ. Such an m was added to the s1 first λ 1 parts. The operation then inserted parts of sizes divisible by m, once s1