A combinatorial problem involving monomial ideals
type: Item
instance of: scholarly article
title: A combinatorial problem involving monomial ideals
zbMATH Open document ID: 0854.13011
DOI: 10.1016/0022-4049(94)00131-3
published in: Journal of Pure and Applied Algebra
publication date: 1997-01-06T00:00:00Z
review text: For an integer \(d\) define \(S(d)\) to be the set of all triples \((e_1, e_2, e_3)\in \mathbb{N}^3\) such that \(e_1+e_2+e_3=d\). If \(d\geq 1\), this set clearly can be covered by (i.e. written as the union of) 3-element sets \((e_1, e_2, e_3)+\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}\) originated by triples \((e_1, e_2, e_3)\in S(d- 1)\). Define \(\rho_3 (d)\) to be the minimal number of such sets needed to cover \(S(d)\). Theorem. \(\rho_3 (1)=1\), \(\rho_3 (2)=3\), \(\rho_3 (8)=18\), and, for \(d\neq 1, 2, 8\), \(\rho_3 (d)=\lfloor {1\over 3} {{d+4} \choose 2} \rfloor-3\). Via an obvious isomorphism the main theorem answers for \(n=3\) a question of \textit{A. V. Geramita}, \textit{D. Gregory} and \textit{L. Roberts} [J. Pure Appl. Algebra 40, 33-62 (1986; Zbl 0586.13015) and Rend. Semin. Mat., Torino 42, No. 2, 1-10 (1984; Zbl 0584.14001)] asking for the smallest number of monic monomials of degree \(d-1\) in \(k[ x_1, \dots, x_n]\) such that the ideal they generate contains all monomials of degree \(d\). The theorem is established by hard elementary combinatorial analysis (many case distinctions, etc.) yielding for all \(d\geq 12\) the estimate \(\rho_3 (d)\geq \rho_3 (d-3)+d+2\). The converse estimate then, following easily from the papers by \textit{A. V. Geramita}, \textit{D. Gregory} and \textit{L. Roberts} cited above, leads to a recursion with above solution. Reviewer's remarks: (a) The result reminds of essentially polynomial solutions to other enumerative problems involving \(S(d)\). The elegant ways in which these can be obtained -- see, e.g., in \textit{R. P. Stanley}'s book ``Enumerative combinatorics. I'' (Monterey 1986; Zbl 0608.05001) exc. 6, ch. 4 -- seem, however, inadaptable. (b) Be aware of possible typoes: e.g., line 5 of the proof of lemma 2.1 should begin with: \dots thus \((0, e_2+1, e_3+1)\not\in \dots\;\).
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 825332
zbMATH Keywords: smallest number of monic generating monomials
zbMATH Keywords: combinatorial analysis
author: Frank Curtis
reviewed by: Alexander Kovačec
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00131-3
OpenAlex ID: W2133811822
cites work: Monomial ideals and points in projective space
cites work: Q5522742
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description: scientific article; zbMATH DE number 825332