The endomorphism of Grassmann graphs

A graph is called a pseudo-core if every endomorphism is either an automorphism or a colouring. In this paper, we show that every Grassmann graph $J_q(n,m)$ is a pseudo-core. Moreover, the Grassmann graph $J_q(n,m)$ is a core whenever $m$ and $n-m+1$ are not relatively prime, and $J_q(2pk-2, pk-1)$ is a core whenever $p,k\geq 2$.Comment: 8 page

Mahonian Pairs

We introduce the notion of a Mahonian pair. Consider the set, P^*, of all words having the positive integers as alphabet. Given finite subsets S,T of P^*, we say that (S,T) is a Mahonian pair if the distribution of the major index, maj, over S is the same as the distribution of the inversion number, inv, over T. So the well-known fact that maj and inv are equidistributed over the symmetric group, S_n, can be expressed by saying that (S_n,S_n) is a Mahonian pair. We investigate various Mahonian pairs (S,T) with S different from T. Our principal tool is Foata's fundamental bijection f: P^* -> P^* since it has the property that maj w = inv f(w) for any word w. We consider various families of words associated with Catalan and Fibonacci numbers. We show that, when restricted to words in {1,2}^*, f transforms familiar statistics on words into natural statistics on integer partitions such as the size of the Durfee square. The Rogers-Ramanujan identities, the Catalan triangle, and various q-analogues also make an appearance. We generalize the definition of Mahonian pairs to infinite sets and use this as a tool to connect a partition bijection of Corteel-Savage-Venkatraman with the Greene-Kleitman decomposition of a Boolean algebra into symmetric chains. We close with comments about future work and open problems.Comment: Minor changes suggested by the referees and updated status of the problem of finding new Mahonian pairs; [email protected] and [email protected]

A new formula for the coefficients of Gaussian polynomials

We deduce exact integral formulae for the coefficients of Gaussian, multinomial and Catalan polynomials. The method used by the authors in the papers [2, 3, 4] to prove some new results concerning cyclotomic and polygonal polynomials, as well as some of their extensions is applied.O. Bagdasar’s research was supported by a grant of the Romanian National Authority for Research and Innovation, CNCS/CCCDI UEFISCDI, project number PN-III-P2-2.1-BG-2016-0333, within PNCDI III

Factors of the Gaussian coefficients

AbstractWe present some simple observations on factors of the q-binomial coefficients, the q-Catalan numbers, and the q-multinomial coefficients. Writing the Gaussian coefficient with numerator n and denominator k in a form such that 2k⩽n by the symmetry in k, we show that this coefficient has at least k factors. Some divisibility results of Andrews, Brunetti and Del Lungo are also discussed