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

For any uncovered set S of patterns, fracR_S,nn converges in distribution to a Bernoulli random variable Ber(p) for some p ∈ [0, 1]. If S contains a pattern starting with 1, then p = 1, and if S contains a pattern starting with its largest…

The set of two-pop-stack sortable permutations of length 2n+1 with exactly n ascents has an equal number of permutations with last block of size one as permutations with last block size greater than one. That is, a(2n+1, n) = 2b(2n+1, n)…

It is possible to adapt Theorem 1 to arbitrary posets (instead of 0, 1-posets) and to injective maps φ with x < y ⇔ φ(x) < φ(y) (instead of x ≤ y ⇔ φ(x) ≤ φ(y))?

We have max_t m_H_t(2, ∞) = max_t m_G_t(2, ∞) and max_t m_H_t(-∞, -2) = max_t m_G_t(-∞, -2).

What is the cop number of the flower snark J_n ?

We believe that results similar to Theorem 4.3.5 can be proved for intersection graphs of other scalable objects. In particular, we conjecture that similar techniques apply to intersection graphs of (unit) regular hexagons.

For all k ≥ 0, the grid of size 3 × (2k+1) is an N-position for the game PUSH-CRAM. In addition, it is a bluff game.

For a simple example, consider the MINIMUM VERTEX COVER problem in fractionally treewidth-fragile graphs, or more generally in hereditary classes with sublinear separators. While the unweighted version can be dealt with by the local search…

Thus it remains to determine h_T(C_n) when n ≡ 1 or 2 (mod 3).

Take an n × n determinant such that its first column is the column of integers in the sequence of sequences S_i rectangular array and its first row is the k^th row, k=1,2,3,… . Then its determinant is given by (∏_j=1^n+1j^j-2)·( cn+k-1 n ).

Let r and q be integers such that 2 ≤ r ≤ 9 and r ≤ q. Then for all n ≥ r, the Turán graph T_r(n) has more q-colorings than any other graph with the same number of vertices and edges.

For k,n,α ∈N , let G ∈ G_n,α have the minimum spectral radius in G_n,α . Then for sufficiently large n, (1) G ≅ F(k,k,k+1) for α=3,n=3k+1, (2) G ≅ F(k+1,k,k+1) for α=3 and n=3k+2, (3) G ≅ F(k,k,k,k+1) for α=4 and n=4k+1, (4) G ≅…

For every triangle-free graph G of minimum degree d, fracα(G)overlineα_G(1)≥ 2-o_d(1).

We have not been able to prove either way.

A balanced magic square of order 5 is completely balanced.

Is there a function f(x, y) such that for any graphs G_1 and G_2 we have χ_{td}(G_1 □ G_2) ≤ f(χ_{td}(G_1), χ_{td}(G_2))?

Is the reflexive dimension of the Minkowski sum P + P' bounded by refldim(P) + refldim(P') + c?

We conjecture that in the case of skips of j and j+1 we have in fact equality, and not just a lower bound using K-groupings.

For any nonnegative integer k, does there exist a graph G that satisfies φ_(2,j)(G) − κ_(2,j)(G) + 1 = k?

What is the exact asymptotics of ex_e(G_1, n)?