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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 i-th 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 Self-Organized 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 167-169.

LINKS

Robert Israel, Table of n, a(n) for n = 1..2000

S. Corteel and D. Gouyou-Beauchamps, Enumeration of Sand Piles, Discrete Maths, 256 (2002) 3, 625-643.

D. Dhar, P. Ruelle, S. Sen and D. Verma, Algebraic aspects of sandpile models, Journal of Physics A 28: 805-831, 1995.

E. Goles and M.A. Kiwi, Games on line graphs and sand piles, Theoretical Computer Science 115: 321-349, 1993

M. Latapy, R. Mantaci, M. Morvan and H. D. Phan, Structure of some sand piles model, Theoret. Comput. Sci. 262 (2001), 525-556.

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/(1-x^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/(1-x^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/(1-x^k)), {k, n}], {n, max}] + O[x]^max; CoefficientList[gf, x] (* Jean-Fran├žois 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

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Last modified November 21 00:06 EST 2019. Contains 329348 sequences. (Running on oeis4.)