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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.

We conjecture that i_k(n)/i_k-1(n) is a decreasing function of k for any n.

Characterize non-König-Egerváry graphs satisfying; • varrho_e(G) ≥ m(G) - ξ(G) + ε(G); • varrho_v(G) = n(G) - ξ(G) + ε(G); • varrho_v(G) = α(G) + μ(G) - ξ(G) + ε(G).

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?

After a big number of experiments the author conjectured that for all finite Abelian groups G,all homomorphisms f:G^3→ G and all periodic initial conditions g,h:N→ G , the resulting recurrent double sequence can also be generated by an…

Let t_k,q(n) be defined by (5.1) t_k,q(j) = q^j, for 0 ≤ j ≤ k-1 t_k,q(n) = q ∑_l=0^k-2 (q-1)^l t_k,q(n-(l+2)), for n ≥ k. Then, S_F_q(T_2,3,…,k(n)) = t_k,q(n) for all values of n ≥ k.

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?

Find other possible values of the parameter d and the corresponding d-antimagic labeling of type (1, 1, 1) for the generalized Petersen graph P(n, 2).

Characterize the (m, n)-extremal graphs for all m and n.

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

We believe the assertion of Theorem 1.6 is true even for all smaller values of n, though we don't have a proof yet.