Problems

Let n ≥2 k+1, k ≥1. Let D_k be the set of permutations of S_n with k descents. Let A_k be the set of permutations with k ascents.There is a bijection σ:D_k→ A_k which satisfies σ(x)≥x in weak Bruhat ordering.

For a pr-graph H,B_H has a special bipartite-min ordering if and only if it has a parity-symmetric one.

Conjecture 6.3. We have H_2,1^(k)(x)=frac1+2x^k1-2x+2x^k-2x^k+1.

With what frequencies do each of these 6 cases occur?

However, we conjecture σ_{6}(2^{k}\cdot 3)=8 for all k ∈N.

However, the jury is still out on whether or not any of the 10-point chirotopes are realizable in 3D space.

The question remains, however, is there a transformation G such that G(T(x))= M_{T}(x) ? If so, what is it?

ℓ(7,3)=11.

Is any relation between diam(F_p,q_1∪ F_p,q_2) and diam F_p,q_1+ +diam F_p,q_2 ? Are there any non-trivial p,q_1,q_2 such that these numbers are the same?

For any r ≥ 2 there exists n_0 = n_0(r) such that if F ⊆ 2^[n] is r-wise intersecting with n ≥ n_0, then the number of (r+1)-triangles in F is at most N(Δ_r+1, F_X), where F_X = F ⊆ [n] : |F ∩ X| ≥ |X|-1 for some (r+1)-set X.

Let G be a class 1 regular graph with Δ > n/3. If any graph obtained from G by splitting a vertex is a critical class 2 graph.

Is the double sequence A^1,id vertically C-log-concave?

It is open if this is true for all cycle-free graphs E, which would show that the assumption "S idempotent" can be dropped.

Determine whether μ_s(D_n)=n+1for even integers n with n \ge 6.

(3) χ(G(2,9,7))=4?

While we are not able to prove it, we think it is a very safe conjecture that R_2(m) converges to below 0.607 , as m tends toward infinity.

Based on the values for |X_n| for small values of n we conjecture that |X_n| = o(n), and leave open the question of enumerating the members of X_n in ascending order, in O(|X_n|) time.

SM_{P_{h}}(P_{n})=\frac{4nh-3n+h-(lh+th-l^{2}-t^{2}-t)}{2} for any integer 2 \le h \le n and 1 \le l \le h, 1 \le t \le h-1 such that n \equiv l (\bmod h) and n-1 \equiv t (\bmod (h-1)).

We do not succeed to solve the case D which is left as an open question (for this case, values in Table 1 are experimentally obtained).

Does there exist a finite t_0 such that d_t(H)=d_t_0(H) for t≥t_0?