Lattice Paths and Rogers Identities

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 combinato-rial identities.


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 "  q  n t  -color partitions" or partitions with " copies of ".Using these new partitions many more -identities have been interpreted combinatorially in [8][9][10][11][12].
Recently in [13] we interpreted combinatorially the following -identities of Rogers [14] by using colored partitions: q q q q q q q q q q q q n n n n n q q q q q q q q q q q q q q q q q q q q q q q q q q        , (1.3)  , , , ; n n n n n q q q q q q q q q q q q q q         , (1.4) and q q q q q q q q q q q q q q In Equations (1.1)-(1.5),n is a rising -factorial which in general is defined as follows : and and   , , , ; = ; .
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 " n t  copies of " (also called  n  n t  -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: , , , .
n t n n n   Thus, for example, the partitions of 2 with " 1 n  copies of " are n , 2 0 , 1 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 to 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 0 3 9 12 17 = 41.
    Recently in [13] we showed that the identities (1.1)-(1.5)have their colored partition theoretic interpretations in the following theorems, respectively:   A  denote the number ofcolor partitions of n  such that even parts appear with even subscripts and odd with odd , if i is the smallest or the only part in the partition, then and the weighted difference of any two consecutive parts is nonnegative and is A  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 2 i is a part and the weighted difference of any two consecutive parts is nonnegative and is 5 In [13] we have shown that for 1  the lefthand side of the Equation generates Here we shall prove that the left-hand side of equation 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. ; ; m m m q q q q q the factor q generates the lat- , which are encoded by inserting horizontal steps in front of the first mountain and horizontal steps in front of the   If a 1 = 8, a 2 = 4, a 3 = 4, a 4 = 0, then our above graph becomes: The factor   generates nonnegative multiples . 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.In the Graph-8, we consider two successive peaks, say th and th and denote them by and , respectively i Now, due to the impact of the factor  4 4 1 ; m q q , the Figure 11 changes to Figure 12 Again by taking into consideration, the impact of the factor     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 x A and y B , respectively, then If we look at the -color part n x A , we find that the parity of both A and x is determined by A and x are even and if A and x are odd.This proves that even parts appear with even subscripts and odd with odd.Clearly, all subscripts x are .> 2 The weighted difference of these two consecutive parts is x 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 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 r   > 0 r between the peaks and Q .In this case there is a descent of from 1 Q to V and an ascent of from V to .This implies But since the weighted difference is nonnegative, therefore .0 r  Also, imply that the height of each peak is atleast 3.This completes the proof of Theorem 6.

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 , = 2 k m 4i 1 i m    between ith and   1 i  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.

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

Figure 1 .
Figure 1.Contains five peaks, three valleys, three mountains and one plane.

3 D 4 A 4 C 4 D
the number of partitions of  into parts and  denotes the number of partitions of  into distinct parts  denote the number ofcolor partitions of n  such that even parts appear with even subscripts and odd with odd, all subscripts are , if i is the smallest or the only part in the partition, then and the weighted difference of any two consecutive parts is and  is the number of partitions of  into parts and  denotes the number of partitions of  into distinct parts Then

Figure 3 .
Figure 3. Contains one peak of height fifteen.

Figure 4 .
Figure 4. Contains one plain of length eight and one peak of height seven.

Figure 5 .
Figure 5. Contains one plain of length eight and one peak of height seven.

Figure 6 .
Figure 6.Contains one plain of length eight and one peak of height seven.

Figure 7 . 3 E 4 E 5 E
Figure 7. Contains two peaks of height four, three and one valley at height zero.Theorem 8. Let

Figure 8 .
Figure 8. Contains four peaks each of height three and three valleys each at height zero.

Figure 9 .
Figure 9. Contains two plains each of length four and four peaks each of height three and one valley at height zero.

1 E
 is uniquely generated in this manner.This proves that the L.H.S. of (

Figure 10 .
Figure 10.Contains two plains each of length four and four peaks of height three, five, four, six respectively and one valley at height zero.

Figure 11 .
Figure 11.Contains two peaks of same height.

Figure 12 .
Figure 12.Contains two peaks separated by a plane and length of the plane is a multiple of four.

Figure 13 .
Figure 13.Contains two peaks of which height differs by an odd number and separated by a plane, P2 has more height than P1. 

Figure 14 .
Figure 14.Contains two peaks of which height differs by an odd number and separated by a plane, P1 has more height than P2.To see the reverse implication, we consider two -color parts of a partition enumerated by n   1 E  , say, and .u C

qFigure 15 .
Figure 15.Contains two peaks separated by a plain.

Figure 16 .
Figure 16.Contains two peaks and a valley at height r.
the form will be the colored part corresponding to the first peak. )

2. The weighted difference of two elements
t Definition