List of Problems

1. Bona’s Conjecture on Product of Cycles
2. $u \cdot v = w$ where $u$ and $v$ are long cycles, probability that 1 and 2 are in the same cycle of $w$

3. Conjecture of Thomas Lam on Affine NilCoxeter Algebras
4. http://arxiv.org/pdf/1007.2871v1

5. Pieri rule for toric Schur functions and Affine Stanley symmetric functions
6. http://arxiv.org/pdf/math/0501335v1

7. Let $n\geq 2$ and $t \geq 0$. Let $f(n,t)$ be the number of sequences with $n$ $x$‘s and $2t$ $a_{ij}$‘s, where $1\leq i < j \leq n$, such that each $a_{ij}$ occurs between the i-th $x$ and the j-th $x$ in the sequence. (Thus the total number of terms in each sequence is $n + 2t\binom{n}{2}$.) Then,

$f(n,t) = \frac{(n+tn(n-1))!}{n!t!^n(2t)!^{\binom{n}{2}}} \displaystyle\prod_{j=1}^{n} \frac{((j-1)t)!^2(jt)!}{(1+(n+j-2)t)!}$.

Problem 27 of http://math.mit.edu/~rstan/bij.pdf.

8. The n-cube $C_n$ (as a graph) is the graph with vertex set $\{ 0,1\} ^n$ (i.e., all binary n-tuples), with an edge between u and v if they dif fer in exactly one coordinate. Thus $C_n$ has $2^n$ vertices and $n2^{n-1}$ edges. Then,

$c(C_n) = 2^{2^n-n-1}\displaystyle\prod_{k=1}^{n} k^{\binom{n}{k}}$,

where c(G) is the number of spanning trees of G.

9. Find a bijection between a staircase tableaux with no $\delta$ of size $n$ and perfect matchings of $\{1,2,\ldots,2n+2\}$.

http://math.mit.edu/~rstan/papers/staircase2.pdf

10. Conjectures on Cambrian Lattices and Generalized Associahedra – already resolved.