A note on X-rays of permutations and a problem of Brualdi and Fritscher

The subject of this note is a challenging conjecture about X-rays of permutations which is a special case of a conjecture regarding Skolem sequences. In relation to this, Brualdi and Fritscher [Linear Algebra and its Applications, 2014] posed the following problem: Determine a bijection between extremal Skolem sets and binary Hankel X-rays of permutation matrices. We give such a bijection, along with some related observations. 1. Skolem sequences Skolem sequences originates from the work by Thoralf Skolem in 1957 [8] on the construction of Steiner triple systems. Skolem proved that the set {1, 2, . . . , 2n} can be partitioned in n pairs (si, ti) such that ti − si = i for i = 1, 2, . . . , n, if and only if n ≡ 0, 1 (mod 4). This result can be reformulated as: There is a sequence with two copies of every element k in A = {1, 2, . . . , n} such that the two copies of k are placed k places apart in the sequence, if and only if n ≡ 0, 1 (mod 4). For example, the set {1, 2, 3, 4} can be used to form the sequence 42324311, but the set {1, 2, 3} cannot be used to form such a sequence. For more information on Skolem sequences and generalizations thereof, see the survey [7]. A natural generalization is when the set of differences A is any set or multiset of positive integers. If A is a multiset, the sequences are called multi Skolem sequences, and the corresponding existence question is that of deciding for which multisets A = {a1, . . . , an} there is a partition of {1, . . . , 2n} into the differences in A. A set A such that there is a partition of {1, . . . , 2n} into the differences in A is called a multi Skolem set. In my MSc thesis [4], I identified (rather obvious) parity and density conditions that are necessary for A to be a multi Skolem set. These conditions were far from sufficient (unsurprisingly as the existence question for multi Skolem sets turns out to be NP-complete [6]). But, surprisingly, I discovered that when A is an ordinary set (i.e., not a multiset), then these simple necessary conditions seem to be sufficient. Conjecture 1 ([4]). A set A = {a1, a2, . . . , an} with a1 < a2 < · · · < an is a Skolem set if and only if the number of even ai’s is even, and ∑n i=m ai ≤ n −(m−1) for each 1 ≤ m ≤ n. A particularly interesting special case of Conjecture 1 emerge when the set of differences A = {a1, a2, . . . , an} is as sparse as possible (in the sense that adding 1 to any element in A force A to violate the density condition), i.e., ∑n i=1 ai = n . Such sets (multisets) A = {a1, a2, . . . , an} satisfying ∑n i=1 ai = n 2 are called extremal. Conjecture 2 ([5]). A set A = {a1, a2, . . . , an} with a1 < a2 < · · · < an and ∑n i=1 ai = n 2 is an extremal Skolem set if and only if ∑n i=m ai ≤ n 2 − (m− 1) for each 1 ≤ m ≤ n. Note that the parity condition in Conjecture 1 is implied by ∑n i=1 ai = n , and that the conjecture is invalid for extremal multisets. A minimal counterexample is A = {4, 4, 4, 8, 8, 8}.