Johnson, Keith \(P\)-orderings of finite subsets of Dedekind domains. (English) Zbl 1180.13024 J. Algebr. Comb. 30, No. 2, 233-253 (2009). Let \(R\) be a Dedekind domain, \(P\) a prime ideal of it and \(K\) the quotient field \(R/P\) with the cardinal number \(q=|R/P|\). For \( x\in R\), let \(\gamma (x)\) denotes the largest integer \(k\) for which \(x\in P^{k}\).An ordering of a finite set \(S\) with \(|S|=n\) is a bijection \(\phi :\langle n\rangle \to S\) where \( \langle n\rangle =\{ 1,2,\dots ,n\} ,\) and \( \langle 0\rangle =\varnothing .\)For non negative integers \(k_{1},k_{2},\dots ,k_{q}\) such that \(n=\sum k_{i}.\) A \((k_{1},k_{2},\dots ,k_{q})\)-shuffle is an ordered set of \(q\) strictly increasing maps \(\phi _{j}:\langle k_{i}\rangle \to \langle n\rangle \) with disjoint images. For non negative integers \( k_{1},k_{2},\dots ,k_{q}\) such that \(m\leq \sum k_{i},\) A \( (k_{1},k_{2},\dots ,k_{q};m)\)-alignment is an ordered set of \(q\) strictly increasing maps \(\phi _{j}:\langle k_{i}\rangle \to \langle m\rangle \) the union of whose images is \(\langle m\rangle .\) A \(P\)-ordering of \(S\) is an ordering \(\{ a_{i},i=1,2,\dots ,|S|\} \) of \(|S|\) with the property that for each \(i>1\) the element \(a_{i}\) minimizes \(\gamma \left( \prod_{j<i}(s-a_{i})\right) \) among all elements \(s\) of \(S\). If \(S,S^{\prime }\) are two finite subsets of \(R\) of the same cardinality, \(m\), then a bijection \(\phi :S\to S^{\prime }\) is a \(P\)-ordering equivalence if \(\phi \) preserves \(P\)-orderings.The author of the paper under review uses the concepts of shuffle and alignment to reconstruct \(P\)-ordering of \(S\) from that of some specific subsets of \(S\) to present upper and lower bounds for the number of distinct \( P\)-orderings a finite set can have in terms of its cardinality and also give an upper bound on the number of \(P\)-ordering equivalence classes of a given cardinality. Reviewer: Manouchehr Misaghian (Prairie View) Cited in 8 Documents MSC: 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations Keywords:P-ordering; P-sequence; Dedekind domain × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bhargava, M.: P-orderings and polynomial functions on arbitrary subsets of Dedekind rings. J. Reine Angew. Math. 490, 101-127 (1997) · Zbl 0899.13022 [2] Bhargava, M.: The factorial function and generalizations. Am. Math. Mon. 107, 783-799 (2000) · Zbl 0987.05003 · doi:10.2307/2695734 [3] Boulanger, J., Chabert, J.-L., Evrard, S., Gerboud, G.: The characteristic sequence of integer-valued polynomials on a subset. Lect. Notes Pure Appl. Math. 205, 161-174 (1999) · Zbl 0964.13011 [4] Boulanger, J., Chabert, J.-L.: Asymptotic behavior of characteristic sequences of integer-valued polynomials. J. Number Theory 80, 238-259 (2000) · Zbl 0973.11093 · doi:10.1006/jnth.1999.2432 [5] Comtet, L.: Advanced Combinatorics, the Art of Finite and Infinite Expansions. Riedel, Dordrecht (1974) · Zbl 0283.05001 [6] Diaconis, P., Mathematical developments from the analysis of riffle shuffles, Durham, 2001, River Edge · Zbl 1026.60005 [7] Kaparthi, S., Rao, H.R.: Higher dimensional restricted lattice paths. Discrete Appl. Math. 31(3), 279-289 (1991) · Zbl 0754.05004 · doi:10.1016/0166-218X(91)90055-2 [8] Polya, G.: Uber Ganzwertige Polynome in Algebraischen Zahlkorper. J. Reine Angew. Math. (Crelle) 149, 97-116 (1919) · JFM 47.0163.04 · doi:10.1515/crll.1919.149.97 [9] Torres, A., Cobada, A., Nieto, J.: An exact formula for the number of alignments between two DNA sequences. DNA Seq. 14, 427-430 (2003) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.