We evaluate the probabilities of various events under the uniform distribution on the set of 312-avoiding permutations of 1, . . . , N . We derive exact formulas for the probability that the i th element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312-avoiding permutation has k specified decreasing points, and we show that for large N the points below the diagonal look like trajectories of a random walk.

Let N denote the set of positive integers and Z the set of all integers. Let N 0 = N ∪ {0}. Let a 1 x 2 + b 1 xy + c 1 y 2 and a 2 z 2 + b 2 zt + c 2 t 2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ N 0 as a sum of these two binary quadratic forms isWhen (b 1 , b 2 ) = (0, 0) we prove under certain conditions on a 1 , b 1 , c 1 , a 2 , b 2 and c 2 that N (a 1 , b 1 , c 1 , a 2 , b 2 , c 2 ; n) can be expressed as a finite linear combination of quantities of the type N (a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N (a, 0, b, c, 0, d; n) are known, we can determine N (a 1 , b 1 , c 1 , a 2 , b 2 , c 2 ; n). This determination is carried out explicitly for a number of quaternary quadratic forms a 1 x 2 + b 1 xy + c 1 y 2 + a 2 z 2 + b 2 zt + c 2 t 2 . For example, in Theorem 1.2 we show 1395 Int.

For a given permutation τ , let P τ N be the uniform probability distribution on the set of N -element permutations σ that avoid the pattern τ . For τ = µ k := 123 · · · k, we consider P µ k N (σ I = J) where I ∼ γN and J ∼ δN for γ, δ ∈ (0, 1). If γ + δ = 1 then we are in the large deviations regime with the probability decaying exponentially, and we calculate the limiting value of P µ k N (σ I = J) 1/N . We also observe that for τ = λ k, := 12 . . . k(k − 1) . . . ( + 1) and γ + δ < 1, the limit of P τ N (σ I = J) 1/N is the same as for τ = µ k . arXiv:1606.07906v1 [math.CO]

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In this paper, we derive some new identities satisfied by the series ∞ m,n=−∞ q m 2 +mn+2n 2 using Ramanujan's identities for L(q), M (q) and N(q). Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.