COMBINATORIAL PROOFS OF GENERATING FUNCTION IDENTITIES FOR F-PARTITIONS

Transcription

1 COMBINATORIAL PROOFS OF GENERATING FUNCTION IDENTITIES FOR F-PARTITIONS AE JA YEE 1 Abstract In his memoir in 1984 George E Andrews introduces many general classes of Frobenius partitions (simply F-partitions) Especially he focuses his interest on two general classes of F-partitions: one is F-partitions that allow up to k repetitions of an integer in any row and the other is F-partitions whose parts are taken from k copies of the nonnegative integers The latter are called k colored F-partitions or F-partitions with k colors Andrews derives the generating functions of the number of F-partitions with k repetitions and F-partitions with k colors of n and leaves their purely combinatorial proofs as open problems The primary goal of this article is to provide combinatorial proofs in answer to Andrews request 1 Introduction For a nonnegative integer n a generalized Frobenius partition or simply an F-partition of n is a two-rowed array of nonnegative integers a1 a 2 a r b 1 b 2 b r where each row is of the same length each is arranged in nonincreasing order and n r+ r (a i+b i ) Frobenius studied F-partitions [4] in his work on group representation theory under the additional assumptions a 1 > a 2 > > a r 0 and b 1 > b 2 > > b r 0 By considering the Ferrers graph of an ordinary partition we see that a i form rows to the right of the diagonal and b i form columns below the diagonal For example the F-partition for is as is seen easily from the Ferrers graph in Figure 1 Thus F-partitions are another representation of ordinary partitions Figure 1 Ferrers graph of In his memoir [2] George E Andrews introduces many general classes of F-partitions Especially he focuses his interest on two general classes of F-partitions: one is F-partitions that allow up to k repetitions of an integer in any row and the other is F-partitions whose parts are taken from k copies of the nonnegative integers The latter are called k colored F-partitions or F-partitions with k colors 1 Research partially supported by a grant from the Number Theory Foundation 1

2 2 AE JA YEE Andrews [2] derives the generating functions of the number of F-partitions with k repetitions and F-partitions with k colors of n and leaves their purely combinatorial proofs as open problems More precisely he [2 p40] offers ten series of problems; in this paper we offer solutions to two series of problems comprising a total of five separate problems The primary goal of this article is to provide combinatorial proofs in answer to Andrews request In Section 2 we interpret the enumerative proof of Jacobi s triple product of E M Wright in another way using the generating function for two-rowed arrays of nonnegative integers In Section 3 we prove the generating function for F-partitions with k colors in a combinatorial way which was independently and earlier established by F G Garvan [6] and in Section 4 we prove identities arising in the study on F-partitions with k colors of Andrews We prove the generating function for F-partitions with k repetitions combinatorially in Section 5 2 Wright s enumerative proof of Jacobi s triple product The well-known Jacobi triple product identity is z n q n2 (1 q 2n )(1 + zq 2n 1 )(1 + z 1 q 2n 1 ) (21) n n1 for z 0 q < 1 There are two proofs of Andrews [1 2] A proof of Wright [8] is combinatorial and involves a direct bijection of bipartite partitions Proofs due to Cheema [3] and to Sudler [7] are variations of Wright s proof Here we describe Wright s proof which we use in the following sections By substituting zq for z and q for q 2 in (21) and then dividing by n1 (1 qn ) on both sides we obtain z n q n(n+1)/2 n (1 + zq n )(1 + z 1 q n 1 ) (22) (1 q n ) n1 n1 which we use below Let A be the set of two-rowed arrays of nonnegative integers u1 u 2 u s (u; v) : v 1 v 2 v t where u 1 > u 2 > > u s 0 v 1 > v 2 > > v t 0 and s and t can be distinct We define the weight of (u; v) A by s t (u; v) : s + u i + v i When z 1 the right side of (22) is the generating function for (u; v) A q (u;v) (u;v) A Thus we interpret Wright s 1-1 correspondence using A graphical representation of a two-rowed array u1 u 2 u s v 1 v 2 v t To do that we first need to describe a that is analogous to the Ferrers graph of an ordinary partition Put s circles on the diagonal and then put u j nodes in row j to the right of the diagonal and v j nodes in column (s t + j) below the diagonal The diagram of a two-rowed array coincides with the Ferrers graph when the lengths of the two rows are equal Figure 2 shows the diagram of

3 Combinatorial Proofs of Generating Function Identities for F-partitions 3 Figure 2 We are ready to construct Wright s 1-1 correspondence Let r s t Suppose that r 0 For the given diagram of (u; v) we separate the diagram into two sets of nodes along the vertical line between column r and column r + 1 to obtain the Ferrers graph of an ordinary partition and a right angled triangle of nodes consisting of r rows Meanwhile if r < 0 then we separate the diagram along the horizontal line between row 0 and row 1 The triangle of nodes consists of (r + 1) rows Note that the number of the nodes in the triangle is r(r + 1)/2 in both cases The ordinary partitions and triangles contribute to n1 (1 qn ) 1 and r zr q r(r+1)/2 respectively on the left side of (22) Clearly the process can be carried out in reverse and gives us a bijection to explain (22) For example we consider the diagram of which splits into the partition and the triangle consisting of 2 rows in Figure 3 Figure 3 3 Generating function of F-partitions with k colors We begin this section by defining F-partitions with k colors We consider k copies of the nonnegative integers written j i where j 0 and 1 i k We call j i an integer j in color i and we use the order j i < l h j < l or j l and i < h For a nonnegative integer n an F-partition of n with k colors is an F-partition of n into nonnegative integers with k colors For example when k is an F-partition of 11 with 4 colors since 5 + ( ) + ( ) 11 Let λ be an F-partition with k colors of N (a1 ) a 2 a r b 1 b 2 b r For convenience we call a j and b j the parts of λ We define two statistics Let A i : A i (λ) : # of a j in color i

4 4 AE JA YEE and B i : B i (λ) : # of b j in color i for i 1 2 k Let cφ k (n) be the number of F-partitions with k colors of n and define the generating function CΦ k (q) of cφ k (n) by CΦ k (q) cφ k (n)q n for q < 1 n0 In Section 5 in [2] CΦ k (q) is given in the following theorem Theorem 31 (Theorem 52 Andrews [2]) For q < 1 where CΦ k (q) q Q(m1m2m k 1) m 1m 2m k 1 (31) (1 q n ) k n1 k 1 Q(m 1 m 2 m k 1 ) m 2 i + 1 i<j k 1 m i m j (32) Proof To show (31) we establish a bijection between F-partitions with k colors and (k 1) tuples of integers and k ordinary partitions Let m i : A i (λ) B i (λ) for i 1 2 k We first consider the case when k 2 The identity (31) gives us CΦ 2 (q) q m2 m (33) (1 q n ) 2 n1 Note that A 1 +A 2 B 1 +B 2 and m 1 m 2 We split λ according to the colors of the parts: λ µ+ν where µ and ν are the two-rowed arrays consisting of parts of λ in colors 1 and 2 respectively and each row of µ and λ is arranged in nonincreasing order Say α1 α 2 α A1 γ1 γ 2 γ A2 µ and ν β 1 β 2 β B1 δ 1 δ 2 δ B2 We apply Wright s 1-1 correspondence described in Section 2 to µ and ν We consider the diagrams of µ and ν By symmetry it is sufficient to consider the case when m 1 0 Since m 1 0 the number of the parts of µ in the first row is greater than or equal to the number of the parts of µ in the second row Thus µ has a triangle to the left of its diagram and ν has a triangle at the top of its diagram We obtain two ordinary partitions by separating the nodes and circles in the triangle from the diagrams of µ and ν Since the triangle from µ has m 1 (m 1 + 1)/2 of nodes and circles and the triangle from ν has m 1 (m 1 1)/2 nodes we see that the sum of the weights of the two partitions newly obtained is N m 2 1 as required The process can be carried out in reverse and gives us a bijection between the set of F-partitions with 2 colors and the set of triples of an integer and two ordinary partitions We consider the general case Let λ be an F-partition of N with k colors and split λ according to the colors of the parts: λ λ 1 + λ λ k where some λ i may be empty By applying Wright s process to each λ i we obtain k ordinary partitions The number of the nodes and circles in the k triangles is k m i (m i + 1) 2

5 Combinatorial Proofs of Generating Function Identities for F-partitions 5 Since k m i 0 by the definition of m i the number of the nodes and circles in the k triangles is Q(m 1 m 2 m k 1 ) as desired which completes the proof Garvan in his PhD thesis [6 pp 55 68] established a combinatorial proof of Theorem 31 and 4 F-partitions with k colors and q-series In the sequel we use the customary notation for q-series n 1 (a) 0 : (a; q) 0 : 1 (a) n : (a; q) n : (1 aq k ) if n 1 k0 (a; q) : lim n (a; q) n q < 1 Theorem 41 (Theorem 82 Andrews [2]) For q < 1 where CΦ k (q) 1 (q) n 1 n k 1 0 m 1 m k 1 0 k 1 R(n 1 n k 1 m 1 m k 1 ) (n 2 i + m 2 i ) + q R(n 1n k 1 m 1 m k 1 ) (q) n1 (q) nk 1 (q) m1 (q) mk 1 1 i<j k 1 Proof Let λ be an F-partition with k colors of N a1 a 2 a r b 1 b 2 b r (n i n j + m i m j ) 1 ij k 1 n i m j (41) Let n i A i (λ) and m i B i (λ) for 1 i k Then (n n k ) (m m k ) 0 To obtain the generating function for F-partitions with k colors we consider the generating function for two-rowed arrays made of parts of λ in color i i 1 k 1 ai1 a i2 a ini b i1 b i2 b imi by splitting λ according to colors and then we combine the generating functions together Note that the parts a ij are distinct and so are the b ij Since the generating function for partitions into n distinct parts is q n(n 1)/2 /(q) n we see that the generating function for pairs of partitions consisting of a ij and b ij is q n i(n i 1)/2+m i (m i 1)/2 (q) ni (q) mi However since we need to count the nodes of λ on the diagonal the generating function for two-rowed arrays made of parts of λ in color i i 1 k 1 is q n i(n i +1)/2+m i (m i 1)/2 (q) ni (q) mi Meanwhile for color k we obtain n k 0 q n k(n k +1)/2+m k (m k 1)/2 (q) nk (q) mk q(nk mk)(nk mk+1)/2 (q)

6 6 AE JA YEE by applying Wright s 1-1 correspondence for a fixed m k Thus k CΦ k (q) (q) ni (q) mi n i m i 0 (n 1 + +n k ) (m 1 + +m k )0 n 1 n k m 1 m k 0 (n 1 + +n k ) (m 1 + +m k )0 1 (q) n 1n k m 1m k 0 (n 1+ +n k ) (m 1+ +m k )0 q n i(n i +1)/2+m i (m i 1)/2 q n 1(n 1 +1)/2+ +n k (n k +1)/2+m 1 (m 1 1)/2+ +m k (m k 1)/2 (q) n1 (q) nk (q) m1 (q) mk q P k 1 (ni(ni+1)/2+mi(mi 1)/2)+(n k m k )(n k m k +1)/2 (q) n1 (q) nk 1 (q) m1 (q) mk 1 (42) Since (n n k ) (m m k ) 0 by substituting (n n k 1 ) (m 1 + m k 1 ) for n k m k we see that the the exponent of q in the numerator of each term on the right side of (42) is R(n 1 n k 1 m 1 m k 1 ) which completes the proof In our proofs of Theorems 31 and 41 we have answered Andrews request for combinatorial proofs Andrews also asked for combinatorial proofs of the next two corollaries Indeed our proofs below are perhaps the combinatorial proofs which Andrews sought Corollary 42 (Corollary 82 Andrews [2]) For q < 1 q q n2 +m 2 s2 nm s (43) (q) n (q) m (q) nm 0 Proof By symmetry it suffices to consider the case when n m On the left side of (43) q n2 nm /(q) n counts the partitions with part n as the largest part at least (n m) times and q m2 /(q) m counts the partitions with part m as the largest part at least m times Thus the coefficient of q N on the left side of (43) is the number of partitions of N + nm with the first Durfee square n 2 and the second Durfee square m 2 Meanwhile the right side of (43) is the generating function for pairs of partitions and squares For a given partition of N + nm we change the Ferrers graph by pushing area b up and switching areas c and d to obtain a partition of N + nm (n m) 2 and a square (n m) 2 Since the n m n m n m m d b a c m b n c a n m d process is reversible we complete the proof Andrews noted that some nice combinatorial aspects of the previous corollary are presented in [5] We consider the generalization of Corollary 42 Corollary 43 (Corollary 81 Andrews [2]) For q < 1 n 1 n k 1 0 m 1 m k 1 0 q R(n 1n k 1 m 1 m k 1 ) (q) n1 (q) nk 1 (q) m1 (q) mk 1 q Q(s1s k 1) s 1s k 1 (q) k 1

7 Combinatorial Proofs of Generating Function Identities for F-partitions 7 where R(n 1 n k 1 m 1 m k 1 ) and Q(s 1 s k 1 ) are defined in (41) and (32) Proof We examine R(n 1 n k 1 m 1 m k 1 ) By definition R(n 1 n k 1 m 1 m k 1 ) k 1 (n 2 i + m 2 i n i m i ) + k 1 (n 2 i + m 2 i m i n i ) + 1 i<j k 1 1 i<j k 1 (n i n j + m i m j n i m j n j m i ) (n i m i )(n j m j ) By applying the argument we used in the proof of Corollary 42 we see that q s2 i q n2 i +m2 i mini s i (q) ni (q) mi (q) where s i n i m i Thus n im i 0 q R(n 1n k 1 m 1 m k 1 ) n 1n k 1 0 m 1m k 1 0 n 1n k 1 0 m 1m k 1 0 s 1s k 1 which completes the proof (q) n1 (q) nk 1 (q) m1 (q) nk 1 q P k 1 (n2 i +m2 i nimi)+p 1 i<j k 1 (ni mi)(nj mj) (q) n1 (q) nk 1 (q) m1 (q) nk 1 q P k 1 s2 i +P 1 i<j k 1 sisj (q) k 1 q Q(s 1s k 1 ) s 1 s k 1 (q) k 1 In fact Corollary 42 can be directly generalized for any k Corollary 44 For q < 1 n 1 n k 0 m 1 m k 0 q P k (n2 i +m2 i n im i ) (q) n1 (q) nk (q) m1 (q) mk q s s2 k s 1 s k (q) k 5 F-partitions with k repetitions and q-series In this section we consider F-partitions that allow up to k repetitions of an integer in any row For a nonnegative integer n an F-partition with k repetitions is an F-partition of n into nonnegative integers with k repetitions in any row For example is an F-partition of 12 with 3 repetitions since 5 + ( ) + ( ) 12 Let φ k (n) be the number of F-partitions of n with k repetitions and define the generating function Φ k (q) of φ k (n) by Φ k (q) φ k (n)q n for q < 1 (51) n0

8 8 AE JA YEE In Section 8 in [2] Andrews gives a generating function Φ k (q) for F-partitions with k repetitions in the following theorem Theorem 51 (Theorem 81 Andrews [2]) For q < 1 Φ k (q) rst 0 ( 1) r+s q (k+1)r(r+1)/2+(k+1)s(s 1)/2+t (q k+1 ; q k+1 ) r (q k+1 ; q k+1 ) s (q) t (q) (r s)(k+1)+t (52) Proof We first need to consider F-partitions whose parts are allowed to repeat infinitely many times and then use the principle of inclusion and exclusion for F-partitions with at most k repetitions allowed Let RP be the set of F-partitions with repetitions in parts ie the set of two-rowed arrays of nonnegtive integers a1 a 2 a m b 1 b 2 b m where a 1 a 2 a m 0 and b 1 b 2 b m 0 Since the generating function for ordinary partitions into nonnegative parts is 1 (q) m m0 we see that the generating function for F-partitions with repetitions is q m (q) m (q) m m0 where the numerator q m comes from the m nodes on the diagonal in the Ferrers graph of ( a 1 a 2 a m ) b 1 b 2 b m Let RP(1; 0) be the set of F-partitions with at least one part a appearing more than k times in the top row By breaking up the partition into the k + 1 a s and the remaining parts we can write the partition as ( a1 a 2 a m k 1 a + + a }{{} b 1 b 2 b m k+1 times where the a i s do not necessarily coincide with the old a i s Recall that there are m nodes on the diagonal in the Ferrers graph of ( a 1 a 2 a m ) b 1 b 2 b m Then we see that the generating function for RP(1; 0) is q (k+1) (1 q k+1 ) t0 q t (q) t (q) t+k+1 On the other hand let RP(0; 1) be the set of F-partitions with at least one part b appearing more than k times in the bottom row By breaking up the partition into the k + 1 b s and the remaining parts we can write the partition as a1 a 2 a m b + + b }{{} b 1 b 2 b m k 1 k+1 times where the b i s do not necessarily coincide with the old b i s Then we see that the generating function for RP(0; 1) is 1 q t (1 q k+1 ) (q) t (q) t+k+1 t0 In general for nonnegative integers r and s let RP(r; s) be the set of F-partitions with at least r and s distinct parts appearing more than k times in the top and bottom row respectively Since the generating function for partitions with exactly n 1 distinct parts with each distinct part repeated n 2 times is q n 2n 1 (n 1 +1)/2 (q n2 ; q n2 ) n1 )

9 Combinatorial Proofs of Generating Function Identities for F-partitions 9 we see that the sum of the generating functions for the F-partitions in RP(r; s) is q (k+1)(r(r+1)/2+s(s 1)/2) q t (q k+1 ; q k+1 ) r (q k+1 ; q k+1 ) s (q) t (q) (r s)(k+1)+t Now we apply the principle of inclusion and exclusion to obtain Φ k (q) Thus Φ k (q) ( 1) r+s q (k+1)(r(r+1)/2+s(s 1)/2) q t (q k+1 ; q k+1 ) r (q k+1 ; q k+1 ) s (q) t (q) (r s)(k+1)+t rs 0 which completes the proof t0 t0 References [1] G E Andrews A simple proof of Jacobi s Triple Product Identity Proceedings of the Amer Math Soc 16 (1965) [2] G E Andrews Generalized Frobenius partitions Mem Amer Math Soc 49 No 301 (1984) [3] M S Cheema Vector partitions and combinatorial identities Math Comp 18 (1964) [4] G Frobenius Ü ber die Charaktere der symmetrischen Gruppe Sitzber Preuss Akad Berlin [5] A Garsia and J Remmel A combinatorial view of Andrews proof of the L-M-W conjectures European J Combinatorics 6 (1985) [6] F G Garvan Generalizations of Dyson s Rank PhD Thesis Pennsylvania State University 1986 [7] C Sudler Two enumerative proofs of an identity of Jacobi Proc Edinburgh Math Soc 15 (1966) [8] E M Wright An enumerative proof of an identity of Jacobi J London Math Soc 40 (1965) Department of Mathematics University of Illinois 1409 West Green Street Urbana IL USA address: yee@mathuiucedu