395 research outputs found
Combinatorial Identities for Incomplete Tribonacci Polynomials
The incomplete tribonacci polynomials, denoted by T_n^{(s)}(x), generalize
the usual tribonacci polynomials T_n(x) and were introduced in [10], where
several algebraic identities were shown. In this paper, we provide a
combinatorial interpretation for T_n^{(s)}(x) in terms of weighted linear
tilings involving three types of tiles. This allows one not only to supply
combinatorial proofs of the identities for T_n^{(s)}(x) appearing in [10] but
also to derive additional identities. In the final section, we provide a
formula for the ordinary generating function of the sequence T_n^{(s)}(x) for a
fixed s, which was requested in [10]. Our derivation is combinatorial in nature
and makes use of an identity relating T_n^{(s)}(x) to T_n(x)
Proofs of some binomial identities using the method of last squares
We give combinatorial proofs for some identities involving binomial sums that
have no closed form.Comment: 8 pages, 16 figure
Congruence successions in compositions
A \emph{composition} is a sequence of positive integers, called \emph{parts},
having a fixed sum. By an \emph{-congruence succession}, we will mean a pair
of adjacent parts and within a composition such that . Here, we consider the problem of counting the compositions of
size according to the number of -congruence successions, extending
recent results concerning successions on subsets and permutations. A general
formula is obtained, which reduces in the limiting case to the known generating
function formula for the number of Carlitz compositions. Special attention is
paid to the case , where further enumerative results may be obtained by
means of combinatorial arguments. Finally, an asymptotic estimate is provided
for the number of compositions of size having no -congruence
successions
Restricted ascent sequences and Catalan numbers
Ascent sequences are those consisting of non-negative integers in which the
size of each letter is restricted by the number of ascents preceding it and
have been shown to be equinumerous with the (2+2)-free posets of the same size.
Furthermore, connections to a variety of other combinatorial structures,
including set partitions, permutations, and certain integer matrices, have been
made. In this paper, we identify all members of the (4,4)-Wilf equivalence
class for ascent sequences corresponding to the Catalan number
C_n=\frac{1}{n+1}\binom{2n}{n}. This extends recent work concerning avoidance
of a single pattern and provides apparently new combinatorial interpretations
for C_n. In several cases, the subset of the class consisting of those members
having exactly m ascents is given by the Narayana number
N_{n,m+1}=\frac{1}{n}\binom{n}{m+1}\binom{n}{m}.Comment: 12 page
On Multiple Pattern Avoiding Set Partitions
We study classes of set partitions determined by the avoidance of multiple
patterns, applying a natural notion of partition containment that has been
introduced by Sagan. We say that two sets S and T of patterns are equivalent if
for each n, the number of partitions of size n avoiding all the members of S is
the same as the number of those that avoid all the members of T.
Our goal is to classify the equivalence classes among two-element pattern
sets of several general types. First, we focus on pairs of patterns
{\sigma,\tau}, where \sigma\ is a pattern of size three with at least two
distinct symbols and \tau\ is an arbitrary pattern of size k that avoids
\sigma. We show that pattern-pairs of this type determine a small number of
equivalence classes; in particular, the classes have on average exponential
size in k. We provide a (sub-exponential) upper bound for the number of
equivalence classes, and provide an explicit formula for the generating
function of all such avoidance classes, showing that in all cases this
generating function is rational.
Next, we study partitions avoiding a pair of patterns of the form
{1212,\tau}, where \tau\ is an arbitrary pattern. Note that partitions avoiding
1212 are exactly the non-crossing partitions. We provide several general
equivalence criteria for pattern pairs of this type, and show that these
criteria account for all the equivalences observed when \tau\ has size at most
six.
In the last part of the paper, we perform a full classification of the
equivalence classes of all the pairs {\sigma,\tau}, where \sigma\ and \tau\
have size four.Comment: 37 pages. Corrected a typ
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