On *q*-Olivier Functions

Helmut Prodinger and Tuwani A. Tshifhumulo

The John Knopfmacher Centre for Applicable Analysis and Number Theory

University of the Witwatersrand, Private Bag 3, WITS 2050, Johannesburg, South Africa

University of the Witwatersrand, Private Bag 3, WITS 2050, Johannesburg, South Africa

helmut@maths.wits.ac.za
tat@univen.ac.za

Annals of Combinatorics 6 (2) p.181-194 June, 2002

Abstract:

We consider words with letters
satisfying an up-up-down pattern like Attaching the (geometric) probability to the
letter *i* (with *p=1-q*), every word gets a probability by assuming
independence of letters. We are interested in the probability that a random
word of length *n* satisfies the up-up-down condition. It turns out
that
one has to consider the 3 residue classes (mod 3) separately; then one
can
compute the associated probability generating function. They turn out to
be *q*-analogues of so called Olivier functions.

References:

1. L. Carlitz, Generating functions for a special class of permutations, Proc. Amer. Math. Soc. 47 (1975) 251¨C256.

2. L. Carlitz and R. Scoville, Enumeration of rises and falls by position, Discrete Math. 5 (1973) 45¨C59.

3. L. Carlitz and R. Scoville, Generating functions for certain types of permutations, J. Combin. Theory, Ser. A 18 (1975) 262¨C275.

4. H. Prodinger, Combinatorics of geometrically distributed random variables: new q-tangent and q-secant numbers, Internat. J. Math. Math. Sci. 24 (2000) 825¨C838.

5. N.J.A. Sloane, The on-line encyclopedia of integer sequences, 2001, published electronically, http://www.research.att.com/njas/sequences/.

6. N.J.A. Sloane and S. Plouffe, The On-Line Encyclopedia of Integer Sequences, Academic Press, 1995.

7. T.A. Tshifhumulo, Combinatorics of geometrically distributed random variables: words and permutations avoiding and satisfying contiguous patterns, Ph.D. Thesis, 2002.