Problems

For k a non-negative integer and s, t ≥ k, the coefficient of x^{2s-k}e^{tx} in f_{s+t-1}(x) is given by [x^{2s-k}e^{tx}] f_{s+t-1}(x) = (-1)^{s} \frac{t^{2s+2t-2k-2}}{2^{s-k}\cdot (s-k)! (t-k)!} \cdot Q_{k}(s,t), where Q_{k}(s,t) is a…

Let (5.3) δ_o(j) = 0 if j is even 1 if j is odd. Define r_k(n) by (5.4) r_k(0) = k r_k(j) = 2^j-δ_o(j)· 2, for 1 ≤ j ≤ k-1 r_k(n) = 2 ∑_l=0^k-2r_k(n-(l+2)), for n ≥ k. Then, S(R_2,3,...,k(n))=r_k(n) for all values of n ≥k.

Conjecture 2. f_(n,n,3),3=q^n+8[ cn+2 1 ]_q[ ln 3 ]_q[ l2 1 ]_q f_(n,4,4),3=q^14[ cn-2 2 ]_q[ ln 1 ]_q[ l6 1 ]_q-q^17frac(1-q^4)(1-q^n-3)^2(1-q^n-2)(1-q)^2(1-q^2)^2

Let G1 and G2 be two graphs that are P4-free and 2K2-free. Then the union of G1 and G2 is perfectly orderable.

In particular, is it true that if the realization |\Gamma| of \Gamma through its direct complex \Delta(\Gamma) is a manifold, then the realization of its partial dual |\Gamma^S| is also a manifold?

Characterize König-Egerváry graphs, where varrho_e(G) = m(G) implies core(G) = ker(G).

Determine the complexity of INDEPENDENT DOMINATION OF DIRECT PRODUCTS

Any collection of sets with no empty Venn regions is splittable.

If L is a set of lines in F_q^3 such that |L|=Ω(q^3) , and such that no plane contains ω(q) lines of L, then |P(L)|≥(1-o(1))q^3 .

The sequence of coefficients of F_n^(a)(x) is unimodal.

Do there exist polynomials f_{1}(\zeta),f_{2}(\zeta),f_{3}(\zeta) and f_{4}(\zeta) ,corresponding to every pair (a, b), satisfying (33), given as in (15), satisfying (16)and (17), such that f_{1}(1)=0,\quad f_{2}(1)^{2}=a^{2},\quad…

Computationally, our results do not imply a better bound on the delay in producing the sequence from Theorem 6 and we leave this as an open problem.

Let n be a positive integer. Put a(n) = max{φ(G) − κ(G) + 1 : |V(G)| = n}. Can we determine a(n)? What is the asymptotic behavior of a(n)?

we conjecture that the number of unimodal permutations of length n whose square avoids the consecutive pattern overline213 ,that is, those that avoid the chain (213,312:overline213) , is equal to 2^n-2+n-1 .

If G_1∈ F_p,q,G_2∈ F_p,q_2,q_1,q_2geqq 1 then d(G_1,G_2)leqq q_1+q_2-2. Under which conditions the equality holds?

Which posets Q allow for a map φ as above?

Nevertheless, we conjecture that their dimension is 2.

If D is a dimatroid and C,D ∈ D , then there exist C',D'∈ D of almost equal size whose union is C ∪ D .

For each vertex v in a graph G of order n ≥ 4, b(G-v)=b(G)+lfloorn/2⌋-2 if and only if G=C_4,P_4,2P_2

Can we find the minors for this property?