# Tagged Questions

The tag has no wiki summary.

386 views

### A $q$-analogue of Foulkes' character related to alternating permutations

My paper "Alternating permutations and symmetric functions" at http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain classes of alternating permutations, such as those whose inverse is ...
117 views

### Special values of continuous q - Hermite polynomials

The continuous $q-$Hermite polynomials are defined by $${H_{n + 1}}(x|q) = 2x{H_n}(x|q) +( {q^n}-1){H_{n - 1}}(x|q)$$ with initial values ${H_{ - 1}}(x|q) = 0$ and ${H_0}(x|q) = 1.$ Cf. e.g. ...
373 views

### What does the $q$-Catalan Numbers count?

I had completed a paper describing the $q$-Catalan numbers, which is the $q$-analog of the Catalan numbers. The $n$-th Catalan numbers can be represented by: $$C_n=\frac{1}{n+1}{2n \choose n}$$ and ...
371 views

1k views

### Is there a q-analog to the braid group?

The braid group $B_n$ on $n$ strands fits into a short exact sequence of groups: $$1 \longrightarrow P_n \longrightarrow B_n \longrightarrow S_n \longrightarrow 1,$$ where $S_n$ is the symmetric ...
1k views

### Curious $q$-analogues

Consider the Fibonacci polynomials $$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j}$$ and the Lucas polynomials L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ...
The generating function $f(z)$ of the Catalan numbers which is characterized by $f(z)=1+zf(z)^2$ is D-finite, or holonomic, i.e. it satisfies a linear differential equation with polynomial ...
Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ derangements of $\{1,2,\dots,n\}$ and that there are \$D_n(q)=(n)_q! \left( ...