An open workspace for mathematical discovery. Check proofs, make comments, form collaborations.

Determining the inversion polynomial for the permutations that avoid this subset of consecutive patterns remains an open question.

Are the properties of being Hamiltonian and planar complete?

For every regular multigraph G, Y_2(G) = Y(G).

The set E_n is a minimal generating set in the weak sense: no proper subset of E_n generates C_n .

For m, r nonnegative integers, P_(m^r)(q,t)=∑_substackλ ⊂(m^r) 2-core(λ)=0f_λ(q^-m,q^r/T;q,t)tildeK_(m^r)-λ(q,t,T;± q^1/2,± t^1/2) and for m, r, n nonnegative integers such that r \leq n, P_(m^r)(x^±;q,t)=∑_substackλ ⊂(m^r)…

For two integers n ≥ 3 and p ≥ 4, f(hatS_p^n) = p^n - fracp^n-18 + fracp^n-2 + … + p8 + 5p/8, if p is even p^n - fracp^n-1 + p^n-2 - 5p + 38, if p is odd.

The case when k is odd was posed as an open problem by Nikiforov [53].

they conjectured that L_n is SEAT if n ≥ 2 even and d ∈ 0, 2;

Let T be a 3-colored (in general, m-colored) tournament not containing 3-colored directed triangles. Must T contain a vertex v such that for every other vertex x of T there exists x rightsquigarrow_mv ? (Or equivalently, must T have a…

It is not known whether there is a finite k-chromatic graph of girth at least g and with \chi_{c}^{s}(G)=2k.

We conjecture that uniqueness of the MCB implies U ≠ ∅.

Equivalently, what is the com putational complexity of EDGE CLIQUE COVER on C_4 -free graphs?

If n ≥ 8, then each tournament of order n contains a double point.

Does there exist a graph Γ of order n and level of symmetry equal to n-k/n for arbitrarily large k ≥2 ?

Let S be the parameter matrix of a k-transversal in a d-uniform r-regular hyper graph G. Then the characteristic polynomial of S is φ(λ)=λ^d-2∏_i=j^d(λ-ξ^jkr), where ξ is a d-th primitive root of unity.

Let λ be a Young diagram. Then [∅, λ] has the CDE property if and only if λ is balanced (of any slope).

For every tree T of order at least 3, overlinemc(T) ≤ 3.

Let m ≥ 3 be an integer, and let Γ be a graph coprime to K_m such that Aut(Γ) ≠ 1. If Γ is connected and R-thin, then (Γ, K_m) is stable.

For k=1,2, there exist P^(5)(k,10-k) -designs of order v if and only if: v ≥ 10-k and v ≡ 1bmod 8 or v ≡ 3bmod 8 .

Find the asymptotics of the maximizing index of E_r(n,k) .