Factors of sums and alternating sums involving binomial coefficients and powers of integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$\sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}},$$ where $\epsilon=\pm 1$.Comment: 14 pages, to appear in Int. J. Number Theor

Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

For all nonnegative integers n, the Franel numbers are defined as $$f_n=\sum_{k=0}^n {n\choose k}^3.$$ 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

Random walk generated by random permutations of {1,2,3, ..., n+1}

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the \text{rise-and-descent} sequences characterizing random permutations $\pi$ of $[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding the end-point $X_n = X^{(n)}_n$ of the trajectory of such a permutation-generated random walk (PGRW) at site $X$, and show that in the limit $n \to \infty$ it converges to a normal distribution with a smaller, compared to the conventional P\'olya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of "turns" of the PGRW trajectories.Comment: text shortened, new results added, appearing in J. Phys.

Omnibus Sequences, Coupon Collection, and Missing Word Counts

An {\it Omnibus Sequence} of length $n$ is one that has each possible "message" of length $k$ embedded in it as a subsequence. We study various properties of Omnibus Sequences in this paper, making connections, whenever possible, to the classical coupon collector problem.Comment: 26 page

Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: $[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]$ (e.g. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). We claim that such matrices behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) holds true for them. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so--called Cartier-Foata matrices. Also, they enter Talalaev's hep-th/0404153 remarkable formulas: $det(\partial_z-L_{Gaudin}(z))$, det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in $Z(U(\hat{gl_n}))$ (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints e.g. in Newton id-s fixed, normal ordering convention turned to standard one, refs. adde

Parasitofauna study of the brown trout, Salmo trutta (Pisces, Teleostei) from Corsica (Mediterranean island) rivers

Corsica is a mediterranean island characterised by a great number of rivers. Salmonides are the main fishes which populate these rivers. Very appreciated by fishermen, Salmonides are represented by three species in the insular hydrographical network, among which an autochthonous species, the brown trout (Salmo trutta). In the present work, we have analysed the parasitofauna of this species. According to our knowledge, this research has never been carried out in Corsica. In a first step, we drew up an inventory of the parasites found in this freshwater fish. In a second step, we studied the differences which appeared in the composition of parasite communities of this species

Parasitofauna study of the brown trout,

Corsica is a mediterranean island characterised by a great number of rivers. Salmonides are the main fishes which populate these rivers. Very appreciated by fishermen, Salmonides are represented by three species in the insular hydrographical network, among which an autochthonous species, the brown trout (Salmo trutta). In the present work, we have analysed the parasitofauna of this species. According to our knowledge, this research has never been carried out in Corsica. In a first step, we drew up an inventory of the parasites found in this freshwater fish. In a second step, we studied the differences which appeared in the composition of parasite communities of this species

Ultrastructure of spermiogenesis and spermatozoon of Leptorhynchoides plagicephalus (Acanthocephala, Palaeacanthocephala), a parasite of the sturgeon Acipenser naccarii (Osteichthyes, Acipenseriformes)

This paper describes the ultrastructure of spermiogenesis and the spermatozoon of Leptorhynchoides plagicephalus, an acanthocephalan parasite of the sturgeon Acipenser naccarii, a species which is under the threat of extinction. At the beginning, spermiogenesis in L. plagicephalus is characterized by the presence of a single centriole in the early spermatid. This centriole generates a flagellum with a 9+0 pattern. Another ultrastructural feature observed during the spermiogenesis of L. plagicephalus is the condensation of chromatin to form an ‘‘intranuclear wall’’. The mature spermatozoon of L. plagicephalus presents a reversed anatomy, as observed in other species of the Acanthocephala. The spermatozoon is divided into two parts: an axoneme and a nucleocytoplasmic derivative. The pattern of spermiogenesis and the ultrastructural organization of the spermatozoon of L. plagicephalus are compared with information available on other acanthocephalan species. The appearance of an ‘‘intranuclear wall’’ observed during the present study represents the first record within the Acanthocephala and is unknown from other animal taxa