Abstract

Recently we interpreted five q-series identities of Rogers combinatorially by using partitions with “n + t copies of n” of Agarwal and Andrews [1]. In this paper we use lattice paths of Agarwal and Bressoud [2] to provide new combinatorial interpretations of the same identities. This results in five new 3-way combinatorial identities.

1. Introduction Definitions and the Main Results

In the literature we find that several -identities such as given in Slater’s compendium [3] have been interpreted combinatorially using ordinary partitions by several authors (for example, see Connor [4], Subbarao [5], Subbarao and Agarwal [6] and Agarwal and Andrews [7]). In the early nineteen eighties Agarwal and Andrews introduced a new class of partitions called “-color partitions” or partitions with “copies of”. Using these new partitions many more -identities have been interpreted combinatorially in [8-12].

Recently in [13] we interpreted combinatorially the following -identities of Rogers [14] by using colored partitions:

(1.1)

(1.2)

(1.3)

(1.4)

and

(1.5)

In Equations (1.1)-(1.5), is a rising -factorial which in general is defined as follows :

If is a positive integer, then obviously

and

and is defined by

We remark that Identities (1.1) and (1.2) were also derived by Bailey [15] and appear in [1], Identities (1.3)-(1.5) are also referred as Rogers-Selberg identities (see [3,14,16]).

In this paper we interpret the left-hand sides of (1.1)- (1.5) as generating functions for certain weighted lattice path functions defined by Agarwal and Bressoud in [17]. First we recall the definitions of the partitions with “copies of” (also called -color partitions) and their weighted difference from [12]:

Definition 1. A partition with “copies of”, is a partition in which a part of size, , can come in different colors denoted by subscripts:

Thus, for example, the partitions of 2 with “copies of” are

Note that zeros are permitted if and only if is greater than or equal to one.

Definition 2. The weighted difference of two elements and, , is defined by and is denoted by.

Next, we recall the following description of lattice paths from [17] which we shall be considering in this paper:

All lattice paths will be of finite length lying in the first quadrant. All paths will begin on the y-axis and terminate on the x-axis. Only three moves are allowed at each step:

northeast: from tosoutheast: from to, only allowed ifhorizontal: from to, only allowed along x-axis.

All lattice paths are either empty or terminate with a southeast step: from to.

In describing lattice paths, we shall use the following terminology:

PEAK: Either a vertex on the y-axis which is followed by a southeast step or a vertex preceded by a northeast step and followed by a southeast step.

VALLEY: A vertex preceded by a southeast step and followed by a northeast step. Note that a southeast step followed by a horizontal step followed by a northeast step does not constitute a valley.

MOUNTAIN: A section of the path which starts on either the xor y-axis, which ends on the x-axis, and which does not touch the x-axis anywhere in between the end points. Every mountain has at least one peak and may have more than one.

PLAIN: A section of path consisting of only horizontal steps which starts either on the y-axis or at a vertex preceded by a southeast step and ends at a vertex followed by a northeast step.

Example: The following path has five peaks, three valleys, three mountains and one plain.

The HEIGHT of a vertex is its y-coordinate. The  Weight of a vertex is its x-coordinate. The WEIGHT OF A PATH is the sum of the weights of its peaks.

In the example given above, there are two peaks of height three and three of height two, two valleys of height one and one of height zero.

The weight of this path is

Recently in [13] we showed that the identities (1.1)- (1.5) have their colored partition theoretic interpretations in the following theorems, respectively:

Theorem 1. Let denote the number of - color partitions of such that even parts appear with even subscripts and odd with odd, all subscripts are greater than 2, if is the smallest or the only part in the partition, then and the weighted difference of any two consecutive parts is nonnegative and is. Let

where is the number of partitions of into parts and denotes the number of partitions of into distinct parts Then

Example. = 6, since the relevant partitions are, , , , ,.

Also,

Theorem 2. Let denote the number of -color partitions of such that even parts appear with even subscripts and odd with odd, if is the smallest or the only part in the partition, then and the weighted difference of any two consecutive parts is and is. Let

where is the number of partitions of into parts and denotes the number of partitions of into distinct parts. Then

Theorem 3. Let denote the number of - color partitions of such that even parts appear with even subscripts and odd with odd, if is the smallest or the only part in the partition, then and the weighted difference of any two consecutive parts is nonnegative and is Let

where is the number of partitions of into parts and denotes the number of partitions of into distinct parts Then

Theorem 4. Let denote the number of - color partitions of such that even parts appear with even subscripts and odd with odd, all subscripts are, if is the smallest or the only part in the partition, then and the weighted difference of any two consecutive parts is and is Let

where is the number of partitions of into parts and denotes the number of partitions of into distinct parts Then

Theorem 5. Let denote the number of partitions of with “copies of” such that the even parts appear with even subscripts and odd with odd, all subscripts are if is the smallest or the only part in the partition, then for some is a part and the weighted difference of any two consecutive parts is nonnegative and is Let

where is the number of partitions of into parts and denotes the number of partitions of into distinct parts Then

In this paper we prove the following combinatorial interpretations of the identities (1.1)-(1.5) in terms of lattice paths:

Theorem 6. Let denote the number of lattice paths of weight which start from, have no valley above height 0, the lengths of the plains, if any, are and the height of each peak is greater than 2. Then

Example., since the relevant lattice paths are:

Theorem 7. Let denote the number of lattice paths of weight which start from, have no valley above height 0, the lengths of the plains are and there is a plain of length between any two peaks. Then

Theorem 8. Let denote the number of lattice paths of weight which start from, have no valley above height 0, the lengths of the plains, if any, is and the height of each peak is greater than 1. Then

Theorem 9. Let denote the number of lattice paths of weight which start from, have no valley above height 0, the height of each peak is, there is a plain of length in the beginning of the path and the lengths of the other plains, if any, are Then

Theorem 10. Let denote the number of lattice paths of weight which start from, have no valley above height 0, the height of each peak is, the lengths of the plains, if any, are Then

Theorems 6-10 lead to the following 3-way extension of Theorems 1-5:

Theorem 11. For, we have

In [13] we have shown that for the lefthand side of the Equation generates and consequently. Here we shall prove that the left-hand side of equation generates also. We shall also show bijectively that

Furthermore, since each of these five cases is proved in a similar way, we provide the details for in our next section and sketch the changes required to treat the remainder in Section 3.

2. Proof of Theorem 6

In the factor generates the lattice path of m peaks each of height 3 starting at (0,0) and terminating at.

If m = 4, the path begins as:The factor generates m-nonnegative multiples of 4, say, which are encoded by inserting horizontal steps in front of the first mountain and horizontal steps in front of the mountain,

If a₁ = 8, a₂ = 4, a₃ = 4, a₄ = 0, then our above graph becomes:

The factor generates nonnegative multiples of (2i1), say,. This is encoded by having the ith peak grow to height

. Each increase by one in the height of a given peak increases its weight by one and the weight of each subsequent peak by two.

If b₁ = 3, b₂ = 1, b₃ = 2, b₄ = 0, then our example becomes:

In the Graph-8, we consider two successive peaks, say th and th and denote them by and, respectively Now, due to the impact of the factor, the Figure 11 changes to Figure 12 Again by taking into consideration, the impact of the factor, the Figure 12 changes to Figure 13 or Figure 14 depending on whether or In the case when, the new graph will look like Figure 12.

Every lattice path enumerated by is uniquely generated in this manner. This proves that the L.H.S. of (1.1) generates.

We now establish a correspondence between the lattice paths enumerated by and the -color partitions enumerated by.

We do this by encoding each path as the sequence of the weights of the peaks with each weight subscripted by the height of the respective peak.

Thus, if we denote the two peaks in Figure 13 (or Figure 14) by and, respectively, then

If we look at the -color part, we find that the parity of both and is determined by. If is odd, then both and are even and if is even, then both and are odd. This proves that even parts appear with even subscripts and odd with odd. Clearly, all subscripts are.

The weighted difference of these two consecutive parts is

Obviously, if is the first peak in the lattice path then it will correspond to the smallest part in the corresponding -color partition or to the singleton part if the -color partition has only one part and in both cases

To see the reverse implication, we consider two -color parts of a partition enumerated by, say, and.

Let and be the corresponding peaks in the associated lattice path.

The length of the plain between the two peaks is which is the weighted difference between the two parts and and is therefore nonnegative and

Also, there can not be a valley above height 0. This can be proved by contradiction.

Suppose, there is a valley V of height between the peaks and.

In this case there is a descent of from to V and an ascent of from V to. This implies

But since the weighted difference is nonnegative, therefore.

Also, imply that the height of each peak is atleast 3. This completes the proof of Theorem 6.

3. Sketch of the Proofs of Theorems 7-10

Case is treated in exactly the same manner as the first case except that now the path begins with peaks each of height 1 and with a plain of length,between ith and th peak.

In the Case, the only point of departure from the first case is that the path begins with peaks each of height 2.

Case is treated in exactly the same manner as the previous case except that the extra factor puts a plain of length of 2 in front of the first peak. This increases the weight of each peak by 2 and so the weight of the lattice path is increased by.

Comparing the case with the case, we see that in this case there are two extra factors, viz., and. The extra factor puts two south east steps: (0,2) to (1,1) and (1,1) to (2,0). Thus there are now peaks starting from (0,2) and the extra factor

introduces a nonnegative multiple of, say. This is encoded by having the first peak grow to height. Clearly,which is of the form will be the colored part corresponding to the first peak.

4. Conclusions

The sum-product identities like (1.1) to (1.5) are generally known as Rogers-Ramanujan type identities. They have applications in different areas such as Orthogonal polynomials, Lie-algebras, Combinatorics, Particle physics and Statistical mechanics.

The most obvious question arising from this work is:

Do Theorems 1.6-1.10 admit generalization analogous to the generalized results of [12,17]?

5. References

[1] A. K. Agarwal and G. E. Andrews, “Rogers-Ramanujan Identities for Partitions with ‘Copies of’,” Journal of Combinatorial Theory, Vol. 45, No. 1, 1987, pp. 40-49. doi:10.1016/0097-3165(87)90045-8

[2] A. K. Agarwal and D. M. Bressoud, “Lattice Paths and Multiple Basic Hypergeometric Series,” Pacific Journal of Mathematics, Vol. 136, No. 2, 1989, pp. 209-228.

[3] L. J. Slater, “Further Identities of the Rogers-Ramanujan Type,” Proceedings of the London Mathematical Society, Vol. s2-54, No. 1, 1952, pp. 147-167. doi:10.1112/plms/s2-54.2.147

[4] W. G. Connor, “Partition Theorems Related to Some Identities of Rogers and Watson,” Transactions of the American Mathematical Society, Vol. 214, 1975, pp. 95-111. doi:10.1090/S0002-9947-1975-0414480-9

[5] M. V. Subbarao, “Some Rogers-Ramanujan Type Partition Theorems,” Pacific Journal of Mathematics, Vol. 120, 1985, pp. 431-435.

[6] M. V. Subbarao and A. K. Agarwal, “Further Theorems of the Rogers-Ramanujan Type,” Canadian Mathematical Bulletin, Vol. 31, No. 2, 1988, pp. 210-214. doi:10.4153/CMB-1988-032-3

[7] A. K. Agarwal and G. E. Andrews, “Hook Differences and Lattice Paths,” Journal of Statistical Planning and Inference, Vol. 14, No. 1, 1986, pp. 5-14. doi:10.1016/0378-3758(86)90004-2

[8] A. K. Agarwal, “Partitions with ‘Copies of’, Lecture Notes in Math.,” Proceedings of the Colloque de Combinatoire Énumérative, Université du Québec à Montréal, Berlin, May 28-June 1, 1985, no. 1234, pp. 1-4.

[9] A. K. Agarwal, “Rogers-Ramanujan Identities for nColor Partitions,” Journal of Number Theory, Vol. 28, No. 3, 1988, pp. 299-305. doi:10.1016/0022-314X(88)90045-5

[10] A. K. Agarwal, “New Combinatorial Interpretations of Two Analytic Identities,” Proceedings of the American Mathematical Society, Vol. 107, No. 2, 1989, pp. 561-567. doi:10.1090/S0002-9939-1989-0979216-7

[11] A. K. Agarwal, “New Classes of Infinite 3-Way Partition Identities,” ARS Combinatoria, Vol. 44, 1996, pp. 33-54.

[12] A .K. Agarwal and G. E. Andrews, “Rogers-Ramanujan Identities for Partitions with ‘Copies of’,” Journal of Combinatorial Theory, Series A, Vol. 45, No. 1, 1987, pp. 40-49. doi:10.1016/0097-3165(87)90045-8

[13] M. Goyal and A. K. Agarwal, “Further Rogers-Ramanujan Identities for n-Color Partitions,” Utilitas Methematica, Winnipeg. (In Press)

[14] L. J. Rogers, “Second Memoir on the Expansion of Certain Infinite Products,” Proceedings of the London Mathematical Society, Vol. s1-25, No. 1, 1894, pp. 318-343. doi:10.1112/plms/s1-25.1.318

[15] W. N. Bailey, “Some Identities in Combinatory Analysis,” Proceedings of the London Mathematical Society, Vol. s2-49, No. 1, 1947, pp. 421-435. doi:10.1112/plms/s2-49.6.421

[16] A. Selberg, “Über Einige Arithmetische Identitäten,” Avhandlinger Norske Akad, Vol. 8, 1936, pp. 1-23.

[17] A. K. Agarwal and D. M. Bressoud, “Lattice Paths and Multiple Basic Hypergeometric Series,” Pacific Journal of Mathematics, Vol. 136, No. 2, 1989, pp. 209-228.