A generalized Mahonian statistic on absorption ring mappings
The article under review describes and discusses a statistic on a new combinatorial object, absorption ring mappings. Suppose we are given a \(k\)-tuple of sequences of distinct integers together with a permutation \(\sigma \in S_n\). An absorption ring mapping is obtained by inserting\ those sequences into a circular arrangement of cells (or ring) which is labelled according to \(\sigma \). The actual insertion process begins by choosing an initial placement for the first sequence. The final placement of the first (and subsequent) sequences is determined by a coin-flipping scheme. In the end, the process determines a (not necessarily one-to-one) function from the set of integers which occur in the given sequences to the set of labels, \(\{1,\ldots ,n\}\). The number of tails in the shortest coin-flipping sequence required to generate the function is, therefore, a statistic on the collection of all such functions (absorption ring mappings). Set \textbf{J} to be the given \(k\)-tuple of sequences, \(j_1,\dots,j_k\) the lengths of those sequences, \(AR_n({\mathbf J})\) the collection of absorption ring mappings generated as described above, and, for \(f\in AR_n({\mathbf J})\), let \(|f|\) denote the number of tails in a minimal flipping sequence. The key result gives the distribution generating function for this statistic, that is, \[ \sum_fq^{|f|}=[n][n-j_1]\cdots [n-j_1-j_2-\cdots -j_k] \] irrespective of choice of initial placement rule or choice of \(\sigma \). The proof follows easily from the definitions and the independence of Bernoulli trials. In case \(k=n\), so that each sequence has just one term, the collection of absorption ring mappings is \(S_n\) and the statistic \(|\;|\) reduces to the generalized Mahonian statistic \(den^a\) described by \textit{G.-H. Han} [``Calcul Denertien'', doctoral thesis, IRMA, Univ. de Strasbourg, 1991]. Moreover, with only modest modifications, \(|\;|\) can be viewed as a statistic on multiset permutations; again, it is equivalent to Han's \(den^a\). In both cases the author shows how various choices of initial placement rule and \(\sigma \) lead to known Mahonian statistics (e.g., inversion number, major index, \(r\)-major index, Denert's statistic).
scientific article; zbMATH DE number 1051480