A grammar of Dumont and a theorem of Diaconis-Evans-Graham

Abstract

We came across an unexpected connection between a remarkable grammar of Dumont for the joint distribution of $(\mathrm{exc},\mathrm{fix})$ over $S_n$ and a beautiful theorem of Diaconis-Evans-Graham on successions and fixed points of permutations. With the grammar in hand, we demonstrate the advantage of the grammatical calculus in deriving the generating functions, where the constant property plays a substantial role. On the grounds of left successions of a permutation, we present a grammatical treatment of the joint distribution investigated by Roselle. Moreover, we obtain a left succession analogue of the Diaconis-Evans-Graham theorem, exemplifying the idea of a grammar assisted bijection. The grammatical labelings give rise to an equidistribution of $(\mathrm{jump},\mathrm{des})$ and $(\mathrm{exc},\mathrm{drop})$ restricted to the set of left successions and the set of fixed points, where jump is defined to be the number of ascents minus the number of left successions.