Abstract:
For any coprime integers (a,b) the expression Cat(a,b)_q = [a+b choose b]_q/[a+b]_q --- which is called a rational q-Catalan number --- is known to be a polynomial with positive integer coefficients. The polynomials appear in many places. However, the nature of the coefficients is still not well-understood. We will review what is known about this problem and we will suggest a new point of view using generating functions on lattice points in certain polytopes. We present a specific conjecture that would not only give a satisfying interpretation of Cat(a,b)_q, but would also give a combinatorial proof that Cat(a,c)_q - Cat(a,b)_q has positive coefficients when b<c and gcd(a,b)=gcd(a,c)=1.
Lattice points and rational q-Catalan numbers
08.10.2024 15:00 - 16:30
Organiser:
I. Fischer, M. Schlosser
Location: