On Symmetric Sandpiles

An efficient algorithm for generating symmetric ice piles

“…In [5] and [12] the Symmetric Sand Pile Model SSPM(n) (obtained by closing {(n)} with respect to RFall and LFall) has been studied and its relation with SPM(n) analysed. Here we define the Symmetric Ice Pile Model SIPM k (n) as the set of integer sequences (called symmetric ice piles) obtained by closing {(n)} w. r. t. RFall, RSlide k , LFall and LSlide k .…”

“…A natural extension of SPM(n) has been introduced [5], [12] in order to fix the lack of symmetry (grains either stay or move to the right). Thus, a grain possibly falls down nondeterministically from column i either to column i − 1 or to column i+1.…”

“…While SPM(n) turns out to be a distributive lattice with exactly one fixed point (the bottom, easily characterized), SSPM(n) is neither a lattice nor admits a unique fixed point. Nevertheless, a characterization of fixed points exists [5,Section 3.2], [12,Th. 2] and their number is known [5,Lemmas 15,16,17].…”

“…Nevertheless, a characterization of fixed points exists [5,Section 3.2], [12,Th. 2] and their number is known [5,Lemmas 15,16,17].…”

Piles de sable décroissantes 1D. Classification expérimentale d'émergences

“…Ces grains se déplacent entre les cases du tableau selon des règles d'évolution comme celles décrites par les Figures 1 et 2. Une attention particulière aété portée aux configurations obtenues en partant d'une unique colonne contenant un nombre fini de grains [8,9,11,6,17,26,27,13,14,10,7,16], amenant des développements mathématiques intéressants pour eux-même, et motivés par troiséléments :…”