490 research outputs found
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
False theta functions and companions to Capparelli's identities
Capparelli conjectured two modular identities for partitions whose parts
satisfy certain gap conditions, where were motivated by the calculation of
characters for the standard modules of certain affine Lie algebras and by
vertex operator theory. These identities were subsequently proved and refined
by Andrews, who related them to Jacobi theta functions, and also by
Alladi-Andrews-Gordon, Capparelli, and Tamba-Xie. In this paper we prove two
new companions to Capparelli's identities, where the evaluations are expressed
in terms of Jacobi theta functions and false theta functions.Comment: 17 pages; references update
Crossings and nestings in colored set partitions
Chen, Deng, Du, Stanley, and Yan introduced the notion of -crossings and
-nestings for set partitions, and proved that the sizes of the largest
-crossings and -nestings in the partitions of an -set possess a
symmetric joint distribution. This work considers a generalization of these
results to set partitions whose arcs are labeled by an -element set (which
we call \emph{-colored set partitions}). In this context, a -crossing or
-nesting is a sequence of arcs, all with the same color, which form a
-crossing or -nesting in the usual sense. After showing that the sizes of
the largest crossings and nestings in colored set partitions likewise have a
symmetric joint distribution, we consider several related enumeration problems.
We prove that -colored set partitions with no crossing arcs of the same
color are in bijection with certain paths in \NN^r, generalizing the
correspondence between noncrossing (uncolored) set partitions and 2-Motzkin
paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a
proof that the sequence counting noncrossing 2-colored set partitions is
P-recursive. We also discuss how our methods extend to several variations of
colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further
revised, additional section adde
Polynomial Triangles Revisited
A polynomial triangle is an array whose inputs are the coefficients in
integral powers of a polynomial. Although polynomial coefficients have appeared
in several works, there is no systematic treatise on this topic. In this paper
we plan to fill this gap. We describe some aspects of these arrays, which
generalize similar properties of the binomial coefficients. Some combinatorial
models enumerated by polynomial coefficients, including lattice paths model,
spin chain model and scores in a drawing game, are introduced. Several known
binomial identities are then extended. In addition, we calculate recursively
generating functions of column sequences. Interesting corollaries follow from
these recurrence relations such as new formulae for the Fibonacci numbers and
Hermite polynomials in terms of trinomial coefficients. Finally, properties of
the entropy density function that characterizes polynomial coefficients in the
thermodynamical limit are studied in details.Comment: 24 pages with 1 figure eps include
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