For a directed graphGwith edge set {1, 2, . . . ,k}, a feasible permutations forGis a permutation π on {1, 2, . . . ,k} such that there exists an injective weight functionp: {1, 2, . . . ,k} → Z+for which the sum of the weights of all incoming edges equals the sum of the weights of all outgoing edges at every vertex ofG, provided that ifp(i1)p(i2)p(ik), then π(it) =tfor everyt∈ {1, . . . ,k}. In this paper, we study the number of feasible permutations (denoted byFq,ℓ) for the De Bruijn graphGq,ℓ−1, whereq> 2 is the size of the alphabet. In the caseq= 4, the corresponding De Bruijn graph is used in DNA data storage. To enumerate these feasible permutations, we establish a connection between feasible permutations and regions in a special hyperplane arrangement, denoted byA. Using Zaslavsky’s formula, we present some numerical results and obtain the exact number ofF3,2andF4,2. Since the formula becomes complicated for larger values ofqand ℓ, we concentrate on the case ℓ = 2 and then by counting the regions in a hyperplane sub-arrangement ofA, we provide a lower bound onFq,2. We compare our bound with the latest lower bound onFq,2and show that our bound is Ω(q3J(q)) while the previous one is Ω(q2J(q)), whereJ(q)= (q2−2q+1)!(q2)!/(q2−q)!.

In order to overcome the challenges caused by flash memories and also to protect against errors related to reading information stored in DNA molecules in the shotgun sequencing method, the rank modulation method has been proposed. In the rank modulation framework, codewords are permutations. In this paper, we study the largest size P(n, d) of permutation codes of length n, i.e., subsets of the set Sn of all permutations on {1, ..., n} with the minimum distance at least d ∈ {1, ..., (n/2)} under the Kendall τ-metric. By presenting an algorithm and two theorems, we improve the known lower and upper bounds for P(n, d). In particular, we show that P(n, d) = 4 for all n ≥ 6 and 3/5 (n/2) < d ≤ 2/3 (n/2). Additionally, we prove that for any prime number n and integer r ≤ n/6, P(n, 3) ≤ (n − 1)! − n − 6r/√n2 − 8rn + 20r2 √(n − 1)!/n(n − r)!. This result greatly improves the upper bound of P(n, 3) for all primes n ≥ 37.

We classify the distance-regular Cayley graphs with least eigenvalue $$-2$$ and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.