

A056219


Number of partitions of n in SPM(n): these are the partitions obtained from (n) by iteration of the following transformation: p > p' if p' is a partition (i.e., decreasing) and p' is obtained from p by removing a unit from the ith component of p and adding one to the (i+1)th component, for any i.


7



1, 2, 2, 4, 5, 6, 9, 13, 15, 19, 25, 34, 42, 51, 61, 78, 98, 122, 146, 175, 209, 253, 307, 374, 444, 524, 617, 729, 858, 1016, 1200, 1414, 1649, 1916, 2223, 2586, 2996, 3475, 4031, 4672, 5385, 6191, 7102, 8148, 9329, 10673, 12201, 13957, 15939, 18172
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The SPM (Sand Pile Model) originated in physics, where it is used as a paradigm for SelfOrganized Criticality. Also used in computer science as a model of distributed behavior. It is a special case of Chip Firing Game and more generally it can be viewed as a cellular automaton.
It is known that the sets SPM(n) have a lattice structure. An explicit formula is known for the (unique) fixed point of SPM(n), as well as a characterization of the elements of SPM(n).


REFERENCES

George E. Andrews, Number Theory, Dover Publications, N.Y. 1971, pp 167169.


LINKS

Robert Israel, Table of n, a(n) for n = 1..2000
S. Corteel and D. GouyouBeauchamps, Enumeration of Sand Piles, Discrete Maths, 256 (2002) 3, 625643.
D. Dhar, P. Ruelle, S. Sen and D. Verma, Algebraic aspects of sandpile models, Journal of Physics A 28: 805831, 1995.
E. Goles and M.A. Kiwi, Games on line graphs and sand piles, Theoretical Computer Science 115: 321349, 1993
M. Latapy, R. Mantaci, M. Morvan and H. D. Phan, Structure of some sand piles model, Theoret. Comput. Sci. 262 (2001), 525556.
Index entries for sequences related to cellular automata


FORMULA

Only complicated recursive formulas are known, see Latapy et al.
G.f.: Sum_{n>=1} x^(n*(n+1)/2)*Product_{k=1..n} (x+1/(1x^k)).  Vladeta Jovovic, Jun 09 2007


EXAMPLE

The fifth term of the sequence is 5 since SPM(5) = { (5), (4,1), (3,2), (3,1,1), (2,2,1) }.
The seventh term of the sequence is 9 since SPM(7) = { (7), (6,1), (5,2), (4,3), (5,1,1), (4,2,1), (3,3,1), (3,2,2), (3,2,1,1) }.


MAPLE

N:= 100: # to get a(1) .. a(N)
g:= add(x^(n*(n+1)/2)*mul(x+1/(1x^k), k=1..n), n=1..floor(sqrt(9+8*N)/2)):
S:= series(g, x, N+1):
seq(coeff(S, x, j), j=1..N); # Robert Israel, Oct 20 2016


MATHEMATICA

max = 60; gf = (1/x) Sum[x^(n*(n+1)/2)*Product[(x + 1/(1x^k)), {k, n}], {n, max}] + O[x]^max; CoefficientList[gf, x] (* JeanFrançois Alcover, Oct 20 2016, after Vladeta Jovovic *)


CROSSREFS

Sequence in context: A121269 A211860 A250114 * A085140 A232166 A325555
Adjacent sequences: A056216 A056217 A056218 * A056220 A056221 A056222


KEYWORD

nonn,nice


AUTHOR

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Aug 03 2000


STATUS

approved



