893 research outputs found
The Method of Combinatorial Telescoping
We present a method for proving q-series identities by combinatorial
telescoping, in the sense that one can transform a bijection or a
classification of combinatorial objects into a telescoping relation. We shall
illustrate this method by giving a combinatorial proof of Watson's identity
which implies the Rogers-Ramanujan identities.Comment: 11 pages, 5 figures; to appear in J. Combin. Theory Ser.
Combinatorial Telescoping for an Identity of Andrews on Parity in Partitions
Following the method of combinatorial telescoping for alternating sums given
by Chen, Hou and Mu, we present a combinatorial telescoping approach to
partition identities on sums of positive terms. By giving a classification of
the combinatorial objects corresponding to a sum of positive terms, we
establish bijections that lead a telescoping relation. We illustrate this idea
by giving a combinatorial telescoping relation for a classical identity of
MacMahon. Recently, Andrews posed a problem of finding a combinatorial proof of
an identity on the q-little Jacobi polynomials which was derived based on a
recurrence relation. We find a combinatorial classification of certain triples
of partitions and a sequence of bijections. By the method of cancelation, we
see that there exists an involution for a recurrence relation that implies the
identity of Andrews.Comment: 12 pages, 5 figure
A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape
Tableau sequences of bounded height have been central to the analysis of
k-noncrossing set partitions and matchings. We show here that familes of
sequences that end with a row shape are particularly compelling and lead to
some interesting connections. First, we prove that hesitating tableaux of
height at most two ending with a row shape are counted by Baxter numbers. This
permits us to define three new Baxter classes which, remarkably, do not
obviously possess the antipodal symmetry of other known Baxter classes. We then
conjecture that oscillating tableau of height bounded by k ending in a row are
in bijection with Young tableaux of bounded height 2k. We prove this conjecture
for k at most eight by a generating function analysis. Many of our proofs are
analytic in nature, so there are intriguing combinatorial bijections to be
found.Comment: 10 pages, extended abstrac
Explicit formula for the generating series of diagonal 3D rook paths
Let denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and proof}
of the fact that the generating series admits
the following explicit expression in terms of a Gaussian hypergeometric
function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27
w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire
- …