Some results on superpatterns for preferential arrangements

A superpattern is a string of characters of length n   over [k]={1,2,…,k}[k]={1,2,…,k} that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length k in a certain class. We prove structural and probabilistic results on superpatterns for preferential arrangements  , including (i) a theorem that demonstrates that a string is a superpattern for all preferential arrangements if and only if it is a superpattern for all permutations; and (ii) a result that is reminiscent of a still unresolved conjecture of Alon on the smallest permutation on [n][n] that contains all k-permutations with high probability.