83 research outputs found
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.
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
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
Vicious walkers, friendly walkers and Young tableaux II: With a wall
We derive new results for the number of star and watermelon configurations of
vicious walkers in the presence of an impenetrable wall by showing that these
follow from standard results in the theory of Young tableaux, and combinatorial
descriptions of symmetric functions. For the problem of -friendly walkers,
we derive exact asymptotics for the number of stars and watermelons both in the
absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the
statement of Theorem 4 and its proof were correcte
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Descent classes of permutations with a given number of fixed points
A
desarrangement is a permutation whose first ascent is even. The first author introduced this class of permutations for the purpose of combinatorially explaining a standard recurrence relation for the number of derangements of
n elements. He gave a nice bijection between derangements and desarrangements. Here we extend his bijection to obtain more refined enumeration results relating derangements and desarrangements. In particular, we construct a bijection between descent classes of derangements and descent classes of desarrangements. The following monotonicity result is a consequence of this correspondence:
In any (nontrivial) descent class the number of permutations with exactly k fixed points decreases as k increases
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