Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2018: 0.53

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 17, Issue 1

Issues

Dominance order on signed integer partitions

Cinzia Bisi
  • Corresponding author
  • Dipartimento di Matematica e Informatica, Universitá di Ferrara, Via Machiavelli 35, 44121, Ferrara, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Giampiero Chiaselotti
  • Dipartimento di Matematica e Informatica, Universitá della Calabria, Via Pietro Bucci, Cubo 30b, 87036 Arcavacata di Rende (CS), Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Tommaso Gentile
  • Dipartimento di Matematica e Informatica, Universitá della Calabria, Via Pietro Bucci, Cubo 30b, 87036 Arcavacata di Rende (CS), Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Paolo Antonio Oliverio
  • Dipartimento di Matematica e Informatica, Universitá della Calabria, Via Pietro Bucci, Cubo 30b, 87036 Arcavacata di Rende (CS), Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-02-16 | DOI: https://doi.org/10.1515/advgeom-2016-0033

Abstract

In 1973 Brylawski introduced and studied in detail the dominance partial order on the set Par(m) of all integer partitions of a fixed positive integer m. As it is well known, the dominance order is one of the most important partial orders on the finite set Par(m). Therefore it is very natural to ask how it changes if we allow the summands of an integer partition to take also negative values. In such a case, m can be an arbitrary integer and Par(m) becomes an infinite set. In this paperwe extend the classical dominance order in this more general case. In particular, we consider the resulting lattice Par(m) as an infinite increasing union on n of a sequence of finite lattices O(m, n). The lattice O(m, n) can be considered a generalization of the Brylawski lattice. We study in detail the lattice structure of O(m, n).

Keywords: Number partitions; dominance order

MSC 2010: 05A17; 06A20

Communicated by: G. Gentili

References

  • [1]

    J. A. Aledo, S. Martínez, F. L. Pelayo, J. C. Valverde, Parallel discrete dynamical systems on maxterm and minterm Boolean functions. Math. Comput. Modelling 55 (2012), 666–671. MR2887406 Zbl 1255.37003Google Scholar

  • [2]

    J. A. Aledo, S. Martínez, J. C. Valverde, Parallel dynamical systems over directed dependency graphs. Appl. Math. Comput. 219 (2012), 1114–1119. MR2981304 Zbl 1291.37018Google Scholar

  • [3]

    J. A. Aledo, S. Martinez, J. C. Valverde, Parallel discrete dynamical systems on independent local functions. J. Comput. Appl. Math. 237 (2013), 335–339. MR2966909 Zbl 1248.37077Google Scholar

  • [4]

    J. A. Aledo, S. Martinez, J. C. Valverde, Parallel dynamical systems over special digraph classes. Int.J. Comput. Math. 90 (2013), 2039–2048. Zbl 1329.68177Google Scholar

  • [5]

    J. A. Aledo, S. Martinez, J. C. Valverde, Updating method for the computation of orbits in parallel and sequential dynamical systems. Int. J. Comput. Math. 90 (2013), 1796–1808. MR3171863 Zbl 06297065Google Scholar

  • [6]

    G. E. Andrews, Euler’s “De Partitio numerorum”. Bull. Amer. Math. Soc. (N.S.) 44 (2007), 561–573. MR2338365 Zbl 1172.11031Google Scholar

  • [7]

    D. K. Arrowsmith, C. M. Place, An introduction to dynamical systems. Cambridge Univ. Press 1990. MR1069752 Zbl 0702.58002Google Scholar

  • [8]

    C. Bisi, G. Chiaselotti, A class of lattices and Boolean functions related to the Manickam–Miklös–Singhi conjecture. Adv. Geom. 13 (2013), 1–27. MR3011531 Zbl 1259.05178Google Scholar

  • [9]

    C. Bisi, G. Chiaselotti, G. Marino, P. A. Oliverio, A natural extension of the Young partition lattice. Adv. Geom. 15 (2015), 263–280. MR3365745 Zbl 1317.05018Google Scholar

  • [10]

    C. Bisi, G. Chiaselotti, P. Oliverio, Sand piles models of signed partitions with d piles. ISRN Comb. 2013 (2013), Article ID 615703, 7 pages. Zbl 1264.05007Google Scholar

  • [11]

    C. Bisi, G. Chiaselotti, D. Ciucci, T. Gentile, F. Infusino, Micro and macro models of granular computing induced by the indiscernibility relation. To appear in Inform. Sci. (2017).Google Scholar

  • [12]

    T. Brylawski, The lattice of integer partitions. Discrete Math. 6 (1973), 201–219. MR0325405 Zbl 0283.06003Google Scholar

  • [13]

    G. Cattaneo, M. Comito, D. Bianucci, Sand piles: from physics to cellular automata models. Theoret. Comput. Sci. 436 (2012), 35–53. MR2924735 Zbl 1251.37017Google Scholar

  • [14]

    S. Corteel, D. Gouyou-Beauchamps, Enumeration of sand piles. Discrete Math. 256 (2002), 625–643. MR1935780 Zbl 1013.05010Google Scholar

  • [15]

    E. Goles, M. A. Kiwi, Games on line graphs and sand piles. Theoret. Comput. Sci. 115 (1993), 321–349. MR1224440 Zbl 0785.90120Google Scholar

  • [16]

    É. Goles, M. Latapy, C. Magnien, M. Morvan, H. D. Phan, Sandpile models and lattices: a comprehensive survey. Theoret. Comput. Sci. 322 (2004), 383–407. MR2080235 Zbl 1054.05007Google Scholar

  • [17]

    E. Goles, M. Morvan, H. D. Phan, Sandpiles and order structure of integer partitions. Discrete Appl. Math. 117 (2002), 51–64. MR1881267 Zbl 0998.05005Google Scholar

  • [18]

    J. L. G. Guirao, F. L. Pelayo, J. C. Valverde, Modeling the dynamics of concurrent computing systems. Comput. Math. Appl. 61 (2011), 1402–1406. MR2773412 Zbl 1217.68150Google Scholar

  • [19]

    W. J. Keith, A bijective toolkit for signed partitions. Ann. Comb. 15 (2011), 95–117. MR2785758 Zbl 1233.05031Google Scholar

  • [20]

    M. Latapy, T. H. D. Phan, The lattice of integer partitions and its infinite extension. Discrete Math. 309 (2009), 1357–1367. MR2510543 Zbl 1168.05007Google Scholar

  • [21]

    M. H. Le, T. H. D. Phan, Strict partitions and discrete dynamical systems. Theoret. Comput. Sci. 389 (2007), 82–90. MR2363362 Zbl 1143.68049Google Scholar

  • [22]

    I. G. Macdonald, Symmetric functions and Hall polynomials. Oxford Univ. Press 1995. MR1354144 Zbl 0824.05059Google Scholar

  • [23]

    T. Do, B. Sands, Chop vectors and the lattice of integer partitions. Discrete Math. 312 (2012), 1195–1200. MR2876367 Zbl 1243.05029Google Scholar

  • [24]

    F. L. Pelayo, J. C. Valverde, Notes on “Modeling the dynamics of concurrent computing systems”. Comput. Math. Appl. 64 (2012), 661–663. MR2948612 Zbl 1252.68204Google Scholar

  • [25]

    H. Phan, T. Huong, On the stability of sand piles model. Theor. Comput. Sci. 411 (2010), 594–601. MR2590138 Zbl 1184.68352Google Scholar

  • [26]

    G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). MR0174487 Zbl 0121.02406Google Scholar

  • [27]

    B. E. Sagan, The symmetric group. Springer 2001. MR1824028 Zbl 0964.05070Google Scholar

  • [28]

    R. P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebraic Discrete Methods 1 (1980), 168–184. MR578321 Zbl 0502.05004Google Scholar

  • [29]

    R. P. Stanley, Quotients of Peck posets. Order 1 (1984), 29–34. MR745587 Zbl 0564.06002Google Scholar

  • [30]

    R. P. Stanley, Enumerative combinatorics. Volume 1. Cambridge Univ. Press 2012. MR2868112 Zbl 1247.05003Google Scholar

About the article

Received: 2014-06-20

Revised: 2014-11-02

Published Online: 2017-02-16

Published in Print: 2017-01-01


Citation Information: Advances in Geometry, Volume 17, Issue 1, Pages 5–29, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2016-0033.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
G. CHIASELOTTI, T. GENTILE, and F. INFUSINO
Journal of the Australian Mathematical Society, 2018, Page 1
[2]
Giampiero Chiaselotti, Tommaso Gentile, Federico Infusino, and Paolo A. Oliverio
Journal of Discrete Mathematical Sciences and Cryptography, 2018, Volume 21, Number 5, Page 1069
[3]
G. Chiaselotti, T. Gentile, F. G. Infusino, and F. Tropeano
Discrete Mathematics, Algorithms and Applications, 2018
[4]
G. Chiaselotti and F. Infusino
International Journal of Approximate Reasoning, 2018, Volume 98, Page 25
[5]
G. Chiaselotti, T. Gentile, and F. Infusino
Journal of Algebra and Its Applications, 2018, Page 1950004
[6]
G. Chiaselotti, T. Gentile, and F. Infusino
Information Sciences, 2018
[7]
G. Chiaselotti, T. Gentile, and F. Infusino
ANNALI DELL'UNIVERSITA' DI FERRARA, 2018
[8]
G. Chiaselotti, T. Gentile, F. Infusino, and P.A. Oliverio
Applied Mathematics and Computation, 2018, Volume 320, Page 781
[9]
Giampiero Chiaselotti, Tommaso Gentile, and Federico Infusino
Comptes Rendus Mathematique, 2017
[10]
G. Chiaselotti, T. Gentile, and F.G. Infusino
International Journal of Approximate Reasoning, 2017, Volume 88, Page 333
[11]
Giampiero Chiaselotti, Tommaso Gentile, Federico G. Infusino, and Paolo A. Oliverio
Annali di Matematica Pura ed Applicata (1923 -), 2017, Volume 196, Number 3, Page 1073
[12]
G. Chiaselotti, T. Gentile, and F. Infusino
Knowledge-Based Systems, 2017, Volume 124, Page 144

Comments (0)

Please log in or register to comment.
Log in