The \(r\)-multipermutations
This paper deals with the counting techniques on certain kind of permutations, called \(r\)-multipermutations, on the multiset \(\{1^ r, 2^ r, \dots, n^ r\}\). A multiset is a set with possibly repeated elements and is formally defined as an ordered pair \((S,F)\), \(S\) being a set (say) \(\{m_ 1, \dots, m_ r\}\) and \(f\) a function from \(S\) to the set of nonnegative integers, and is written as \(\{m_ 1^{f(m_ 1)}, \dots, m_ r^{f(m_ r)}\}\). An \(r\)-multipermutation of the set \(\{m_ 1, \dots, m_ n\}\) is a permutation \(a_ 1, a_ 2, \dots, a_{rn}\) of the multiset \(\{m^ r_ 1, m_ 2^ r, \dots, m^ r_ n\}\), where if \(i<j<k\) and \(a_ i = a_ k\) then \(a_ j \geq a_ i\). Rather than considering \(S = \{m_ 1, m_ 2, \dots, m_ n\}\), the set considered here is \(\{1,2, \dots, n\} = [n]\), and the multiset is then \(\{1^ r, 2^ r, \dots, n^ r\} = [n]^{(r)}\). Various interesting results dealing with the counting of such \(r\)-multipermutations by inversions, major index, descents, and left-right minima are derived.
scientific article; zbMATH DE number 617608