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Recently in [1] Goyal and Agarwal interpreted a generalized basic series as a generating function for a colour partition function and a weighted lattice path function. This led to an infinite family of combinatorial identities. Using Frobenius partitions, we in this paper extend the result of [1] and obtain an infinite family of 3-way combinatorial identities. We illustrate by an example that our main result has a potential of yielding Rogers-Ramanujan-MacMahon type identities with convolution property.

A series involving factors like rising

is called basic series (or

Remark: Obviously,

if

The following two “sum-product” basic series identities are known as Rogers-Ramanujan identities:

and

They were first discovered by Rogers [

Theorem (*). The number of partitions of

Theorem (**). The number of partitions of

Partition theoretic interpretations of many more q-series identities like (*) and (**) have been given by several mathematicians (see, for instance, Gӧllnitz [

In all these results, ordinary partitions were used. In [

was interpreted as generating function of two different combinatorial objects, viz., an n-colour partition function and a weighted lattice path function. This led to an infinte family of combinatorial identities. Our objective here is to extend the main result of [

Definition 1.1 A partition with “

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

Definition 1.2 The weighted difference of two elements

Definition 1.3 A two-rowed array of non-negative integers,

with each row aligned in non-increasing order is called a generalized Frobenius partition or more simply an F-partition of ν,

if

Next, we recall the following description of lattice paths from [

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

Northeast: from

Southeast: from

Horizontal: from

The following terminology will be used in describing lattice paths:

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

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.

For the related graphs the reader is referred to the following papers.

(a) T. Mansour, Counting peaks at height k in a Dyck path, Journal of Integer Sequences, 5 (2002), Article 02.1.1.

(b) T. Mansour, Statistics on Dyck paths, Journal of Integer Sequences 9:1 (2006), Article 06.1.5.

(c) P. Peart and W.J. Woan, Dyck paths with no peaks at height k, J. of Integer Sequences 4 (2001), Article 01.1.3.

(d) D. Merlini, R. Sprugnoli and M.C. Verri, Some statistics on Dyck paths, J. Statist. Plann. and Infer. 101 2002, 211-227.

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

In this example, 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

The following result was proved in [

Theorem 1.1 For a positive integer k, let

(1.1a) the parts are of the form

(1.1b) if

(1.1c) the weighted difference between any two consecutive parts is nonnegative and is

Let

(1.1d) they have no valley above height 0

(1.1e) there is a plain of length

(1.1f) the height of each peak of odd (resp., even) weight is 1 (resp. 2) if k is odd and 2 (resp. 1) if k is even. Then

and

In this paper we propose to prove the following theorems which extend Theorem 1.1 for odd and even k separately:

Theorem 1.2 For k an odd positive integer, let

(1.2c)

(1.2d)

As in Theorem 1.1, let

(1.2e) the parts are

(1.2f) only the first copy of the odd parts and the second copy of the even parts are used, that is, the parts are of the form

(1.2g) if

(1.2h) the weighted difference of any two consecutive parts is non-negative and is

Then

Theorem 1.3 For k an even positive integer, let

(1.3c)

(1.3d)

As in Theorem 1.1, let

(1.3e) the parts are

(1.3f) only the second copy of the odd parts and the first copy of the even parts are used, that is, the parts are of the form

(1.3g) if

(1.3h) the weighted difference of any two consecutive parts is non-negative and is

Then

In our next section we give the detailed proof of Theorem 1.2. The proofs of Theorem 1.2 and Theorem 1.3 are similar and hence the proof of Theorem 1.3 is omitted. The interested reader can supply it or obtain from the authors. In Section 3 we illustrate by an example that our results have the potential of yielding Rogers- Ramanujan-MacMahon type combinatorial identities.

We establish a one-one correspondence between the F-partitions enumerated by

enumerated by

colour partition. The mapping

and the inverse mapping

Clearly (2.1) and (1.2a) imply (1.2e). Also (2.1) and (1.2b) imply (1.2f). Since

which is non-negative and is divisible by 4 in view of (1.2d) and so (1.2h) follows.

To see the reverse implication we note that (2.2) and (1.2e) imply (1.2a).

(2.2) and (1.2f) imply (1.2b). Since if

Now suppose

Then in view of (2.2.),we have

Clearly (2.3) and (1.2h) imply (1.2d).

This completes the proof of Theorem 1.2.

To illustrate the bijection we have constructed, we give an example for

Thus

By a little series manipulation, the following identity of Slater [

can be written in the following form:

Now an appeal to Theorem 1.1 gives the following 3-way combinatorial interpretation of Identity (3.2)

Theorem 3.1: Let

where

Example.

Also

For

Our Theorem 1.2 provides a four way extension of (3.3) as follows:

Frobenius partitions enumerated by | Images under |
---|---|

Partitions enumerated by | Partitions enumerated by | Partitions enumerated by | ||||
---|---|---|---|---|---|---|

0 | 1 | Empty partition | 1 | Empty partition | 1 | Empty partition |

1 | 1 | 1_{1} | 0 | - | 1 | 1 |

2 | 1 | 2_{2} | 1 | 2 | 2 | 2, 1 + 1 |

3 | 0 | - | 1 | 3 | 2 | 2 + 1, 1 + 1 + 1 |

4 | 1 | 3_{1} + 1_{1} | 1 | 2 + 2 | 4 | 4, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1 |

5 | 2 | 5_{1}, 4_{2} + 1_{1} | 1 | 3 + 2 | 5 | 5, 4 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 |

6 | 1 | 6_{2} | 3 | 6, 3 + 3, 2 + 2 + 2 | 7 | 5 + 1, 4 + 2, 4 + 1 + 1, 2 + 2 + 2, 2 + 2 + 1 + 1, 2 + 1 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1 + 1 |

7 | 1 | 5_{1 }+ 2_{2} | 1 | 3 + 2 + 2 | 9 | 7, 5 + 2, 5 + 1 + 1, 4 + 2 + 1, 4 + 1 + 1 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1 |

0 | 1 | Empty lattice path with weight 0 |

1 | 1 | |

2 | 1 | |

3 | 0 | - |

4 | 1 | |

5 | 2 | |

6 | 1 | |

7 | 1 |

Frobenius partitions enumerated by | ||
---|---|---|

0 | 1 | Empty Frobenius partition with No column |

1 | 1 | |

2 | 1 | |

3 | 0 | No partition |

4 | 1 | |

5 | 2 | |

6 | 1 | |

7 | 1 |

The work done in this paper shows a nice interaction between the theory of basic series and combinatorics. Theorems 1.1, 1.2 and 1.3 give a 3-way combinatorial identity for each value of k. Thus we get infinitely many 3-way combinatorial identities from these theorems. In one particular case, viz.,

It would be of interest if more applications of Theorems 1.1, 1.2 and 1.3 can be found.