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A new discrete dynamical system of signed integer partitions. (English) Zbl 1333.05026

Summary: T. Brylawski [Discrete Math. 6, 201–219 (1973; Zbl 0283.06003)] described the covering property for the domination order on non-negative integer partitions by means of two rules. Recently, in C. Bisi et al. [“Dominance order on signed partitions”, Adv. Geom. (to appear)], G. Cattaneo et al. [“Non uniform cellular automata description of signed partition versions of ice and sand pile models”, Lect. Notes Comput. Sci. 8751, 115–124, (2014); “The lattice structure of equally extended signed partitions. A generalization of the Brylawski approach to integer partitions with two possible models: ice piles and semiconductors”, Fundam. Inform. 141, 1–36 (2015)] the two classical Brylawski covering rules have been generalized in order to obtain a new lattice structure in the more general signed integer partition context. Moreover, in Cattaneo et al. [loc. cit.], the covering rules of the above signed partition lattice have been interpreted as evolution rules of a discrete dynamical model of a two-dimensional p-n semiconductor junction in which each positive number represents a distribution of holes (positive charges) located in a suitable strip at the left semiconductor of the junction and each negative number a distribution of electrons (negative charges) in a corresponding strip at the right semiconductor of the junction.
In this paper we introduce and study a new sub-model of the above dynamical model, which is constructed by using a single vertical evolution rule. This evolution rule describes the natural annihilation of a hole-electron pair at the boundary region of the two semiconductors. We prove several mathematical properties of such new discrete dynamical model and we provide a discussion of its physical properties.

MSC:

05A17 Combinatorial aspects of partitions of integers

Citations:

Zbl 0283.06003
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References:

[1] Aledo, J. A.; Martínez, S.; Pelayo, F. L.; Valverde, J. C., Parallel discrete dynamical systems on maxterm and minterm Boolean functions, Math. Comput. Modelling, 55, 3-4, 666-671, (2012) · Zbl 1255.37003
[2] Aledo, J. A.; Martínez, S.; Valverde, J. C., Parallel dynamical systems over directed dependency graphs, Appl. Math. Comput., 219, 3, 1114-1119, (2012) · Zbl 1291.37018
[3] Aledo, J. A.; Martínez, S.; Valverde, J. C., Graph dynamical systems with general Boolean states, Appl. Math. Inf. Sci., 9, 4, 1803-1808, (2015)
[4] Aledo, J. A.; Martínez, S.; Valverde, J. C., Parallel dynamical systems over graphs and related topics: A survey, J. Appl. Math., 2015, (2015), Article ID 594294, 14 pages
[5] Andrews, G. E., Euler’s de partitio numerorum, Bull. AMS, 44, 4, 561-573, (2007) · Zbl 1172.11031
[6] Barret, C. L.; Chen, W. Y.C.; Zheng, M. J., Discrete dynamical systems on graphs and Boolean functions, Math. Comput. Simulation, 66, 487-497, (2004) · Zbl 1113.37005
[7] Barret, C. L.; Hunt, H. B.; Marathe, V. M.; Ravi, S. S.; Rosenkrantz, D. J.; Stearns, R. E., On some special classes of sequential dynamical systems, Ann. Comb., 7, 4, 381-408, (2003) · Zbl 1060.68136
[8] Barret, C. L.; Mortveit, H. S.; Reidys, C. M., Elements of a theory of computer simulation. II. sequential dynamical systems, Appl. Math. Comput., 107, 2-3, 121-136, (2000) · Zbl 1049.68149
[9] Barret, C. L.; Mortveit, H. S.; Reidys, C. M., Elements of a theory of computer simulation. III. equivalence of SDS, Appl. Math. Comput., 122, 3, 325-340, (2001) · Zbl 1050.68161
[10] Barret, C. L.; Mortveit, H. S.; Reidys, C. M., Elements of a theory of computer simulation. IV. sequential dynamical systems: fixed points, invertibility and equivalence, Appl. Math. Comput., 134, 1, 153-171, (2003) · Zbl 1028.37010
[11] Barret, C. L.; Reidys, C. M., Elements of a theory of computer simulation. I. sequential CA over random graphs, Appl. Math. Comput., 98, 2-3, 241-259, (1999) · Zbl 0927.68114
[12] Bisi, C.; Chiaselotti, G., A class of lattices and Boolean functions related to the manickam-miklös-singhi conjecture, Adv. Geom., 13, 1, 1-27, (2013) · Zbl 1259.05178
[13] Bisi, C.; Chiaselotti, G.; Gentile, T.; Oliverio, P. A., Dominance order on signed partitions, Adv. Geom., (2016), in press
[14] Bisi, C.; Chiaselotti, G.; Marino, G.; Oliverio, P. A., A natural extension of the Young partition lattice, Adv. Geom., 15, 3, 263-280, (2015) · Zbl 1317.05018
[15] Bisi, C.; Chiaselotti, G.; Oliverio, P. A., Sand piles models of signed partitions with d piles, ISRN Combin., 2013, (2013), Article ID 615703, 7 pages. http://dx.doi.org/10.1155/2013/615703 · Zbl 1264.05007
[16] Brylawski, T., The lattice of integer partitions, Discrete Math., 6, 201-219, (1973) · Zbl 0283.06003
[17] Cattaneo, G.; Chiaselotti, G.; Dennunzio, A.; Formenti, E.; Manzoni, L., Non uniform cellular automata description of signed partition versions of ice and sand pile models, (Cellular Automata, Lecture Notes in Computer Science, vol. 8751, (2014)), 115-124
[18] Cattaneo, G.; Chiaselotti, G.; Gentile, T.; Oliverio, P. A., The lattice structure of equally extended signed partitions. A generalization of the brylawski approach to integer partitions with two possible models: ice piles and semiconductors, Fund. Inform., 141, 1-36, (2015) · Zbl 1344.37015
[19] Cattaneo, G.; Comito, M.; Bianucci, D., Sand piles: from physics to cellular automata models, Theoret. Comput. Sci., 436, 35-53, (2012) · Zbl 1251.37017
[20] Cervelle, J.; Formenti, E., On sand automata, (Mathematical foundations of computer science 2003, Lecture Notes in Computer Science, vol. 2607, (2003), Springer Berlin), 642-653 · Zbl 1035.68065
[21] Cervelle, J.; Formenti, E.; Masson, B., Basic properties for sand automata, (Mathematical foundations of computer science 2005, Lecture Notes in Computer Science, vol. 3618, (2005), Springer Berlin), 192-211 · Zbl 1156.68486
[22] Cervelle, J.; Formenti, E.; Masson, B., From sandpiles to sand automata, Theoret. Comput. Sci., 381, 1-28, (2007) · Zbl 1155.68051
[23] Chiaselotti, G.; Gentile, T.; Oliverio, P. A., Parallel and sequential dynamics of two discrete models of signed integer partitions, Appl. Math. Comput., 232, 1249-1261, (2014) · Zbl 1410.37015
[24] Chiaselotti, G.; Keith, W.; Oliverio, P. A., Two self-dual lattices of signed integer partitions, Appl. Math. Inf. Sci., 8, 3191-3199, (2014)
[25] Cori, R.; Le Borgne, Y., The sand-pile model and Tutte polynomials, Advances in Applied Mathematics, 30, 1-2, 44-52, (2003) · Zbl 1030.05058
[26] Cori, R.; Rossin, D., On the sandpile group of dual graphs, European J. Combin., 21, 4, 447-459, (2000) · Zbl 0969.05034
[27] Corteel, S.; Gouyou-Beauchamps, D., Enumeration of sand piles, Discrete Math., 256, 625-643, (2002) · Zbl 1013.05010
[28] Dennunzio, A.; Guillon, P.; Masson, B., Sand automata as cellular automata, Theoret. Comput. Sci., 410, 38-40, 3962-3974, (2009) · Zbl 1171.68023
[29] Formenti, E.; Masson, B., On computing the fixed points for generalized sandpiles, Int. J. Unconventional Comput., 2, 1, 13-25, (2005)
[30] Formenti, E.; Masson, B., A note on fixed points of generalized ice piles models, Int. J. Unconventional Comput., 2, 2, 183-191, (2006)
[31] Green, C.; Kleitman, D. J., Longest chains in the lattice of integer partitions ordered by majorization, European J. Combin., 7, 1-10, (1986) · Zbl 0605.05003
[32] Holmgren, R. A., A first course in discrete mathematical systems, (1994), Springer-Verlag · Zbl 0797.58001
[33] Keith, W. J., A bijective toolkit for signed partitions, Ann. Comb., 15, 95-117, (2011) · Zbl 1233.05031
[34] Le Borgne, Y.; Rossin, D., On the identity of the sandpile group, Discrete Math., 256, 3, 775-790, (2002) · Zbl 1121.82335
[35] Levine, L., The sandpile group of a tree, European J. Combin., 30, 1026-1035, (2009) · Zbl 1221.05052
[36] Levine, L., Sandpile groups and spanning trees of directed line graphs, J. Combin. Theory Ser. A, 118, 350-364, (2011) · Zbl 1292.05135
[37] Manickam, N.; Singhi, N. M., First distribution invariants and EKR theorems, J. Combin. Theory Ser. A, 48, 91-103, (1988) · Zbl 0645.05023
[38] Mortveit, H. S.; Reidys, C. M., An introduction to sequential dynamical systems, (2008), Springer · Zbl 1004.05056
[39] Reidys, C. M., Sequential dynamical systems over words, Ann. Comb., 10, 4, 481-498, (2006) · Zbl 1130.37334
[40] Reidys, C. M., Combinatorics of sequential dynamical systems, Discrete Math., 308, 4, 514-528, (2008) · Zbl 1128.37006
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