Problems

Consequently one can ask if there is always an optimal circular permutation of the same form as the pattern.

Let P be a finite poset and ω a labeling of P. Then the following two conditions are equivalent. (i) ω is an admissible labeling. (ii) There exists m ∈ N such that ∑_φ ∈ A(P, ω) q^|φ| = q^m ∑_φ ∈ A(P) q^|φ|.

Let G be a tight graph such that: For every edge (u,v)∈ E(G) , one of its endpoints is dense, and the other is non-dense, and |N(u)∩ N(v)|≤ 1 , for all pair of vertices u,v ∈ D(G),u≠v .Then, χ_b(G) = m(G).

In particular, what is the dimension of the restriction of that space to patterns of size k? Is it spanned by corner trees with k vertices?

and for r ≥3 ∑_i=0^n(-1)^n-i( ln i )U_m+k+2+i,k+i^(r)= ll2^n if m=2n, 2^n(5/2n+2r-1) if m=2n+1, . (1.11)

The double sequence A^σ,id assigned to the D'Arcais polynomials is horizontally log-concave.

Conjecture 1.6.2. Let h_i(q) be defined by qh_i(q)=g_i+1,i(q) . Then f_m(b,q)=(1-q)^m-1 +q ∑_i=0^m-1(1-q)^m-y^(i)h_i(q)b^i +∑_i=m^binomm2-1(1-q)^m-y^(i)g_m,i(q)b^i +fracqb^binomm2(m-1)!∑_i=0^m-2⟨ cm-1 i ⟩ q^i.

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.

By using the relation α(G)=ω(Ḡ), can we obtain min(α, r^n), max(α, r^n), Min(α, r^n) and Max(α, r^n)?

Conjecture 5.4. Let w(x;t)=∑_λω(λ)P_λ(x;t), where P_λ(x;t) denote the Hall-Littlewood function corresponding to the partition λ , and the sum runs over all partitions λ . Then rlog w(x;-1)+∑_n ≥ 1 odd1/2na^nc^np_2n+∑_n ≥ 2…

This partial choice of factors has a unique coherent extension to tildeA .

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 .

It seems likely that this is the only such case, though we do not have a proof yet.

More general: does it hold for all monadic-second order definable problems? We conjecture not

lim_l → ∞limsup_n → ∞P_n^av(τ)(A_l;k_n^(n))=0, for all τ ∈ ∪_m=2^∞S_m and for all k_n.

rgirth(F)≤2 ∑_1≤i≤nfrac1|F_i|+1 for any family F=(F_1,...,F_n) of subsets of E(K_n) .

From this, we (very strongly) conjecture that root bifurcating Greg trees have an asymptotic probability of above (and close to) 0.606.

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?

Let h(t) be a polynomial of degree s ≤ d with nonnegative coefficients that satisfies Properties (H) and (S). Does U_r^d+1h(t) have an interlacing symmetric decomposition for all r ≥ d+1 (possibly even for r ≥ maxs, d+1-s)?