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

  5. Pieri rule for toric Schur functions and Affine Stanley symmetric functions

  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

  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\}.

  10. Conjectures on Cambrian Lattices and Generalized Associahedra – already resolved.
    Cambrian Lattices – Nathan Reading
  11. Problem of finding volumes (Ehrhart polynomials) of valuation polytope (EC1 2nd ed. Exer. 4.62)
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