47 research outputs found
A simple proof of Dixon's identity
AbstractWe present another simple proof of Dixon's identity
The Eulerian Distribution on Involutions is Indeed Unimodal
Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point
free involutions) of {1,...,n} with k descents. Motivated by Brenti's
conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is
log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal
in k, for all n. Furthermore, we conjecture that there are nonnegative integers
a_{n,k} such that This statement is stronger than
the unimodality of I_{n,k} but is also interesting in its own right.Comment: 12 pages, minor changes, to appear in J. Combin. Theory Ser.
Factors of binomial sums from the Catalan triangle
By using the Newton interpolation formula, we generalize the recent
identities on the Catalan triangle obtained by Miana and Romero as well as
those of Chen and Chu. We further study divisibility properties of sums of
products of binomial coefficients and an odd power of a natural number. For
example, we prove that for all positive integers ,
, and any nonnegative integer , the expression
is either an integer or a half-integer. Moreover,
several related conjectures are proposed.Comment: 15 pages, final versio
Bijective Proofs of Gould's and Rothe's Identities
We first give a bijective proof of Gould's identity in the model of binary
words. Then we deduce Rothe's identity from Gould's identity again by a
bijection, which also leads to a double-sum extension of the
-Chu-Vandermonde formula.Comment: 4 page
Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas
Motivated by the resemblance of a multivariate series identity and a finite
analogue of Euler's pentagonal number theorem, we study multiple extensions of
the latter formula. In a different direction we derive a common extension of
this multivariate series identity and two formulas of Lucas. Finally we give a
combinatorial proof of Lucas' formulas.Comment: 11 pages, to appear in Discrete Mathematics. See also
http://math.univ-lyon1.fr/~gu
Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers
For all nonnegative integers n, the Franel numbers are defined as We confirm two conjectures of Z.-W. Sun on
congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0
\pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2
\pmod{p^5}, where n is a positive integer and p>3 is a prime.Comment: 8 pages, minor changes, to appear in Integral Transforms Spec. Func
A Generalization of the Ramanujan Polynomials and Plane Trees
Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has
recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1,
Q_{n+1}=[x+nz+(y+t)(n+y\partial_y)]Q_n. In this paper we prove Chapoton's
conjecture on the duality formula: Q_n(x,y,z,t)=Q_n(x+nz+nt,y,-t,-z), and
answer his question about the combinatorial interpretation of Q_n. Actually we
give combinatorial interpretations of these polynomials in terms of plane
trees, half-mobile trees, and forests of plane trees. Our approach also leads
to a general formula that unifies several known results for enumerating trees
and plane trees.Comment: 20 pages, 2 tables, 8 figures, see also
http://math.univ-lyon1.fr/~gu
Some congruences involving central q-binomial coefficients
Motivated by recent works of Sun and Tauraso, we prove some variations on the
Green-Krammer identity involving central q-binomial coefficients, such as where is
the Legendre symbol and is the th cyclotomic polynomial. As
consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q
&\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose
2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for , the
first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence
modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to
appear in Adv. Appl. Mat
Some Arithmetic Properties of the q-Euler Numbers and q-Sali\'e Numbers
For m>n\geq 0 and 1\leq d\leq m, it is shown that the q-Euler number
E_{2m}(q) is congruent to q^{m-n}E_{2n}(q) mod (1+q^d) if and only if m\equiv n
mod d. The q-Sali\'e number S_{2n}(q) is shown to be divisible by
(1+q^{2r+1})^{\left\lfloor \frac{n}{2r+1}\right\rfloor} for any r\geq 0.
Furthermore, similar congruences for the generalized q-Euler numbers are also
obtained, and some conjectures are formulated.Comment: 12 pages, see also http://math.univ-lyon1.fr/~gu
Two truncated identities of Gauss
Two new expansions for partial sums of Gauss' triangular and square numbers
series are given. As a consequence, we derive a family of inequalities for the
overpartition function and for the partition function
counting the partitions of with distinct odd parts. Some further
inequalities for variations of partition function are proposed as conjectures.Comment: 9 pages, final versio