Yee a J - rial Proofs of Generating Function Identities for F-Partitions - J. Comb. Theory Ser.a 102(2003) No.1 217-228

8/9/2019 Yee a J - rial Proofs of Generating Function Identities for F-Partitions - J. Comb. Theory Ser.a 102(2003) No.1 217-228

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Journal of Combinatorial Theory, Series A 102 (2003) 217228

Note

Combinatorial proofs of generating functionidentities for F-partitions

Ae Ja Yee 1

Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USAReceived 15 October 2002

Abstract

In his memoir in 1984, George E. Andrews introduces many general classes of Frobeniuspartitions (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, andthe 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 thegenerating 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 goalof this article is to provide combinatorial proofs in answer to Andrews request.r 2003 Elsevier Science (USA). All rights reserved.

1. Introduction

For a nonnegative integer n; a generalized Frobenius partition or simply anF-partition of n is a two-rowed array of nonnegative integers

a1 a2 ? a rb1 b2 ? br ; !;

where each row is of the same length, each is arranged in nonincreasing order andn r Pri 1a i bi :Frobenius studied F-partitions [4] in his work on group representation theoryunder the additional assumptions a1 4 a24 ? 4 a r X 0 and b1 4 b24 ? 4 br X 0: Byconsidering the Ferrers graph of an ordinary partition, we see that a i form rows to

E-mail address: [email protected] Research partially supported by a grant from the Number Theory Foundation.

0097-3165/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.doi:10.1016/S0097-3165(03)00023-2


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the right of the diagonal and bi form columns below the diagonal. For example, theF-partition for 7+7+5+4+2+2 is

6 5 2 05 4 1 0 !;

as is seen easily from the Ferrers graph in Fig. 1 . Thus F-partitions are anotherrepresentation of ordinary partitions.

In his memoir [2], Andrews introduces many general classes of F-partitions.Especially, he focuses his interest on two general classes of F-partitions: one isF-partitions that allow up to k repetitions of an integer in any row, and the other isF-partitions whose parts are taken from k copies of the nonnegative integers. Thelatter are called k colored F-partitions or F-partitions with k colors. Andrews [2]

derives the generating functions of the number of F-partitions with k repetitions andF-partitions with k colors of n and leaves their purely combinatorial proofs as openproblems. More precisely, he [2, p. 40] offers ten series of problems; in this paper weoffer solutions to two series of problems, comprising a total of ve separateproblems. The primary goal of this article is to provide combinatorial proofs inanswer to Andrews request.

In Section 2, we interpret the enumerative proof of Jacobis 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-partitionswith k colors in a combinatorial way, which was independently and earlierestablished by Garvan [6], and in Section 4, we prove identities arising in the studyon F-partitions with k colors of Andrews. We prove the generating function forF-partitions with k repetitions combinatorially in Section 5.

2. Wrights enumerative proof of Jacobis triple product

The well-known Jacobi triple product identity is

XN

nNznqn

2

YN

n11 q2n1 zq2n11 z1q2n1 2:1

for za 0; jqjo 1: There are two proofs of Andrews [1,2]. A proof of Wright [8] iscombinatorial, and involves a direct bijection of bipartite partitions. Proofs due toCheema [3] and Sudler [7] are variations of Wrights proof. Here we describeWrights proof, which we use in the following sections.

Fig. 1. Ferrers graph of 7 7 5 4 2 2:

Note / Journal of Combinatorial Theory, Series A 102 (2003) 217228218


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By substituting zq for z and q for q2 in (2.1), and then dividing by QNn11 q

nonboth sides, we obtain

PN

nN zn

qnn 1=2

QNn11 qn

YN

n11 zqn1 z1qn1; 2:2

which we use below. Let A be the set of two-rowed arrays of nonnegative integers

u; v :u1 u2 ? usv1 v2 ? vt ; !;

where u14 u2 4 ? 4 usX 0; v14 v2 4 ? 4 vt X 0; and s and t can be distinct. We denethe weight of u; vA A by

ju; vj : s Xs

i 1ui X

t

i 1vi :

When z 1; the right-hand side of (2.2) is the generating function for u; vA A ;

Xu;vA A qju;vj:Thus, we interpret Wrights 1-1 correspondence using A : To do that, we rst need todescribe a graphical representation of a two-rowed array

u1 u2 ? us

v1 v2 ? vt !that is analogous to the Ferrers graph of an ordinary partition. Put s circles on thediagonal, and then put u j nodes in row j to the right of the diagonal and v j nodes incolumn s t j below the diagonal. The diagram of a two-rowed array coincideswith the Ferrers graph when the lengths of the two rows are equal. Fig. 2 shows thediagram of

6 5 2 17 5 4 3 1 0 !:

We are ready to construct Wrights 1-1 correspondence. Let r s t: Supposethat rX 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 theFerrers graph of an ordinary partition and a right angled triangle of nodes consistingof r rows. Meanwhile, if ro 0; then we separate the diagram along the horizontal line

Fig. 2.

Note / Journal of Combinatorial Theory, Series A 102 (2003) 217228 219


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between row 0 and row 1. The triangle of nodes consists of r 1 rows. Notethat the number of the nodes in the triangle is rr 1=2 in both cases. The ordinarypartitions and triangles contribute to

QNn

11 q

n1 and

PNr

N zr qrr 1=2;

respectively, on the left side of (2.2). Clearly, the process can be carried out inreverse, and gives us a bijection to explain (2.2). For example, we consider thediagram of

9 8 5 3 2 1

5 4 1 0 !;which splits into the partition 8 8 6 5 5 5 2 2 and the triangleconsisting of 2 rows in Fig. 3 .

3. Generating function of F-partitions with k colors

We begin this section by dening F-partitions with k colors. We consider k copiesof the nonnegative integers written j i ; where j X 0 and 1 p i p k : We call j i an integer j in color i ; and we use the order

j i o l h 3 j o l ; or j l and i o 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 4;21 13 11 02 0112 11 04 02 01 !

is an F-partition of 11 with 4 colors, since 5 2 1 1 0 0 1 1 0 0 0 11 :

Let l be an F-partition with k colors of N

a1 a2 ? a rb1 b2 ? br

!:

For convenience, we call a j and b j the parts of l : We dene two statistics. Let

Ai : Ai l : # of a j in color i

Fig. 3.



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and

B i : B i l : # of b j in color i

for i 1; 2; y ; k :Let cf k n be the number of F-partitions with k colors of n; and dene the

generating function C F k q of cf k n by

C F k q XN

n0cf k nq

n for jqjo 1:

In Section 5 in [2], C F k q is given in the following theorem.

Theorem 3.1 (Theorem 5.2, Andrews [2]). For jqjo 1;

C F k q PN

m1 ;m2 ;y ;mk 1N qQm1 ;m2;y ;mk 1

QNn1 1 qn

k ; 3:1

where

Qm1; m2; y ; mk 1 Xk 1

i 1m2i X1p i o j p k 1 mi m j : 3:2

Proof. To show (3.1), we establish a bijection between F-partitions with k colors andk 1-tuples of integers and k ordinary partitions. Let

mi : Ai l B i l

for i 1; 2; y ; k :We rst consider the case when k 2: The identity (3.1) gives us

C F 2q PNmN q

m2

QNn1 1 qn

2: 3:3

Note that A1 A2 B 1 B 2 and m1 m2: We split l according to the colors of the parts: l m n; where m and n are the two-rowed arrays consisting of parts of lin colors 1 and 2, respectively, and each row of mand l is arranged in nonincreasingorder. Say

ma1 a2 ? aA1b1 b2 ? bB 1 !and n g1 g2 ? gA2d1 d2 ? dB 2 !:

We apply Wrights 1-1 correspondence described in Section 2 to m and n: Weconsider the diagrams of m and n: By symmetry it is sufcient to consider the casewhen m1X 0: Since m1X 0; the number of the parts of min the rst row is greater thanor equal to the number of the parts of m in the second row. Thus m has a triangle tothe left of its diagram, and n has a triangle at the top of its diagram. We obtain two

ordinary partitions by separating the nodes and circles in the triangle from thediagrams of mand n: Since the triangle from mhas m1m1 1=2 of nodes and circles,and the triangle from n has m1m1 1=2 nodes, we see that the sum of the weights



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of the two partitions newly obtained is N m21 as required. The process can becarried out in reverse, and gives us a bijection between the set of F-partitions with2 colors and the set of triples of an integer and two ordinary partitions.

We consider the general case. Let l be an F-partition of N with k colors, and splitl according to the colors of the parts: l l 1 l 2 ? l k ; where some l i may beempty. By applying Wrights process to each l i ; we obtain k ordinary partitions. Thenumber of the nodes and circles in the k triangles is

Xk

i 1

mi mi 12

:

Since

Pk i 1 mi 0 by the denition of mi ; the number of the nodes and circles in the

k triangles is Qm1; m

2; y ; m

k 1 as desired, which completes the proof. &

Garvan in his Ph.D. thesis [6, pp. 5568] established a combinatorial proof of Theorem 3.1.

4. F-partitions with k colors and q-series

In the sequel, we use the customary notation for q-series

a0 : a ; q0 : 1; an : a ; qn : Yn1

k 01 aq k if nX 1;

and

a ; qN : limn- N a ; qn; jqjo 1:


C F k q 1

qN Xn1 ;y ;nk 1 X 0

Xm1 ;y ;mk 1 X 0

qRn1 ;y ;nk 1;m1 ;y ;mk 1

qn1?

qnk 1 qm1?

qmk 1;

where

Rn1; y ; nk 1; m1; y ; mk 1

Xk 1

i 1n2i m

2i X1p i o j p k 1 ni n j mi m j X1p i ; j p k 1 ni m j : 4:1

Proof. Let l be an F-partition with k colors of N

a1 a2 ? a rb1 b2 ? br !:



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Let ni Ai l and mi B i l for 1 p i p k : Then n1 ? nk m1 ? mk 0: To obtain the generating function for F-partitions with k colors, we consider thegenerating function for two-rowed arrays made of parts of l in color i ; i 1; y ; k 1;

a i 1 a i 2 ? a ini bi 1 bi 2 ? bimi !

by splitting l according to colors, and then we combine the generating functionstogether. Note that the parts a ij are distinct and so are the bij : Since the generatingfunction for partitions into n distinct parts is qnn1=2=qn; we see that the generatingfunction for pairs of partitions consisting of a ij and bij is

qni ni 1=2 mi mi 1=2

qni

qmi

:

However, since we need to count the nodes of l on the diagonal, the generatingfunction for two-rowed arrays made of parts of l in color i ; i 1; y ; k 1; is


qni qmi :

Meanwhile, for color k ; we obtain

Xnk X 0

qnk nk 1=2 mk mk 1=2

qnk qmk

qnk mk nk mk 1=2

qN;

by applying Wrights 1-1 correspondence for a xed mk : Thus,C F k q

Yk

i 1 Xni ;mi X 0n1 ? nk m1 ? mk 0


qni qmi

Xn1 ;y ;nk ;m1 ;y ;mk X 0n1 ? nk m1 ? mk 0

qn1n1 1=2 ? nk nk 1=2 m1m11=2 ? mk mk 1=2

qn1 ? qnk qm1 ? qmk

1

qN Xn1 ;y ;nk ;m1 ;y ;mk X 0n1 ? nk m1 ? mk 0

qPk 1

i 1ni ni 1=2 mi mi 1=2nk mk nk mk 1=2

qn1 ? qnk 1 qm1 ? qmk 1:

4:2

Since n1 ? nk m1 ? mk 0; by substituting n1 ? nk 1 m1 ? mk 1 for nk mk ; we see that the exponent of q in the numerator of each term on the right side of (4.2) is Rn1; y ; nk 1; m1; y ; mk 1; which completesthe proof. &

In our proofs of Theorems 3.1 and 4.1 we have answered Andrews request forcombinatorial proofs. Andrews also asked for combinatorial proofs of the next two



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corollaries. Indeed, our proofs below are perhaps the combinatorial proofs whichAndrews sought.

Corollary 4.2 (Corollary 8.2, Andrews [2]). For jqjo 1;

Xn;mX 0 qn2 m2nm

qnqm P

NsN q

s2

qN: 4:3

Proof. By symmetry, it sufces to consider the case when nX m: On the left-handside of (4.3), qn

2nm=qn counts the partitions with part n as the largest part at least

n m times, and qm2

=qm counts the partitions with part m as the largest part atleast m times. Thus the coefcient of qN on the left-hand side of (4.3) is the numberof partitions of N nm with the rst Durfee square n2 and the second Durfee squarem2: Meanwhile, the right-hand side of (4.3) is the generating function for pairs of partitions and squares. For a given partition of N nm; we change the Ferrers graphby pushing area b up and switching areas c and d to obtain a partition of N nm n m2 and a square n m2: Since the process is reversible, we complete theproof. &

Andrews noted that some nice combinatorial aspects of the previous corollary arepresented in [5]. We consider the generalization of Corollary 4.2.

Corollary 4.3 (Corollary 8.1, Andrews [2]). For jqjo 1;

Xn1;y ;nk 1X 0

Xm1 ;y ;mk 1 X 0

qRn1 ;y ;nk 1 ;m1 ;y ;mk 1

qn1 ? qnk 1 qm1 ? qmk 1

PNs1 ;y ;sk 1N q

Qs1 ;y ;sk 1

qk 1N;



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where R n1; y ; nk 1; m1; y ; mk 1 and Q s1; y ; sk 1 are dened in (4.1) and (3.2).

Proof. We examine Rn1; y ; nk 1; m1; y ; mk 1: By denition,

Rn1; y ; nk 1; m1; y ; mk 1

Xk 1

i 1n2i m

2i ni mi X1p i o j p k 1 ni n j mi m j ni m j n j mi

Xk 1

i 1n2i m

2i mi ni X1p i o j p k 1 ni mi n j m j :

By applying the argument we used in the proof of Corollary 4.2, we see that

Xni ;mi X 0 qn2i m

2i mi ni

qni qmi P

Nsi N q

s2i

qN;

where si ni mi : Thus,

Xn1;y ;nk 1X 0 X

m1 ;y ;mk 1 X 0

qRn1 ;y ;nk 1 ;m1 ;y ;mk 1

qn1 ? qnk 1 qm1 ? qnk 1

Xn1 ;y ;nk 1 X 0 Xm1 ;y ;mk 1 X 0 qPk 1

i 1n2i m

2i ni mi P1p i o j p k 1ni mi n j m j qn1 ? qnk 1 qm1 ? qnk 1

PNs1 ;y ;sk 1N qP

k 1

i 1s2i P1p i o j p k 1 si s j qk 1N

PN

s1 ;y ;sk 1NqQs1;y ;sk 1

qk 1N ;

which completes the proof. &

In fact, Corollary 4.2 can be directly generalized for any k :

Corollary 4.4. For jqjo 1;

Xn1 ;y ;nk X 0 Xm1 ;y ;mk X 0 qPk

i 1n2i m2i ni mi

qn1 ? qnk qm1 ? qmk P

Ns1 ;y ;sk N qs

21 ? s2k

qk N:



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5. F-partitions with k repetitions and q-series

In this section, we consider F-partitions that allow up to k repetitions of an integerin 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,

3 1 1 0 0

1 1 0 0 0 !is an F-partition of 12 with 3 repetitions since 5 3 1 1 0 0 1 1 0 0 0 12: Let f k nbe the number of F-partitions of n with k repetitions, anddene the generating function F k q of f k n by

F k q XN

n0f k nq

nfor jqjo 1: 5:1

In Section 8 in [2], Andrews gives a generating function F k qfor F-partitions with k repetitions in the following theorem.


F k q Xr;s;tX 0 1r sqk 1rr 1=2 k 1ss1=2 t

qk 1; qk 1r qk 1; qk 1sqtqrsk 1t: 5:2

Proof. We rst need to consider F-partitions whose parts are allowed to repeatinnitely 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 withrepetitions in parts, i.e., the set of two-rowed arrays of nonnegative integers

a1 a2 ? amb1 b2 ? bm !;

where a1X a2X ? X amX 0 and b1X b2X ? X bmX 0: Since the generating function

for ordinary partitions into nonnegative parts is

XN

m0

1qm

;

we see that the generating function for F-partitions with repetitions is

XN

m0

qm

qmqm;

where the numerator qm comes from the m nodes on the diagonal in the Ferrers

graph of a1b1

a2b2

?

?

ambm :

Let RP 1; 0 be the set of F-partitions with at least one part a appearing morethan k times in the top row. By breaking up the partition into the k 1 as and the



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remaining parts, we can write the partition as

a ?

a |fflfflfflfflfflfflffl{zfflfflfflfflfflk 1 timesa1a2 ? amk 1

b1b2 ? bm !;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 a1b1a2b2

?

?

ambm : Then, we see that the

generating function for RP 1; 0 is

qk 1

1 qk 1XN

t0

qt

qtqt k 1:

On the other hand, let RP 0; 1 be the set of F-partitions with at least one part bappearing more than k times in the bottom row. By breaking up the partition intothe k 1 bs and the remaining parts, we can write the partition as

b ? b

|fflfflfflfflfflfflffl{zfflfflfflfflfflk 1 timesa1a2 ? am

b1b2 ? bmk 1 !;where the bi s do not necessarily coincide with the old bi s. Then, we see that thegenerating function for RP 0; 1 is

11 qk 1X

N

t0

qt

qtqt k 1:

In general, for nonnegative integers r and s; let RP r; sbe the set of F-partitionswith at least r and s distinct parts appearing more than k times in the top and bottomrow, respectively. Since the generating function for partitions with exactly n1 distinctparts, with each distinct part repeated n2 times, is

qn2n1n1 1=2

qn2 ; qn2 n1;

we see that the sum of the generating functions for the F-partitions in RP r; s is

qk 1rr 1=2 ss1=2

qk 1 ; qk 1r qk 1 ; qk 1s XN

t0

qt

qtqrsk 1 t:

Now, we apply the principle of inclusion and exclusion to obtain F k q: Thus,

F k q

Xr;sX 01r sqk 1rr 1=2 ss1=2

qk 1; qk 1r qk 1; qk 1s XN

t0

qt

qtqrsk 1t;

which completes the proof. &



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References

[1] G.E. Andrews, A simple proof of Jacobis Triple Product Identity, Proceedings of the American

Mathematics Society 16 (1965) 333334.[2] G.E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc. 39 (301) (1984).[3] M.S. Cheema, Vector partitions and combinatorial identities, Math. Comput. 18 (1964) 414420.[4] G. Frobenius, U ber die Charaktere der symmetrischen Gruppe, Sitzber. Preuss. Akad. Berlin, 1900,

pp. 516534.[5] A. Garsia, J. Remmel, A combinatorial view of Andrews proof of the L-M -W conjectures, Eur.

J. Combin. 6 (1985) 335352.[6] F.G. Garvan, Generalizations of Dysons Rank, Ph.D. Thesis, Pennsylvania State University, 1986.[7] C. Sudler, Two enumerative proofs of an identity of Jacobi, Proc. Edinburgh Math. Soc. 15 (1966)

6771.[8] E.M. Wright, An enumerative proof of an identity of Jacobi, J. London Math. Soc. 40 (1965) 5557.


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Yee a J - rial Proofs of Generating Function Identities for F-Partitions - J. Comb. Theory Ser.a 102(2003) No.1 217-228