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Search: id:A056219
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| A056219 |
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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. |
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4
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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).
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Oct 21 2009: (Start)
Product[(1+z*q^n),{n,0Infinity}]=Sum[A[n]*z^n=1+Sum[q^(n(n-1)/2)*z^n/Product[(1-q^k),(k,1,n}],{n,1,Infinity}]
with recursion:
A[n]=q^(n-1)/(1-q^n)*A[n-1]
q^(n(n-1)/2)*z^n/Product[(1-q^k),(k,1,n}]=q^(n*(n + 1))/2*Product[q +1/(1 - q^k), {k, 1, n}], {n, 1, Infinity}]
One can solve for z:
Solve[q^(n(n - 1)/2)*z^n/Product[(1 - q^k), {k, 1, n}] == q^(n*(n + 1))/2*Product[q + 1/(1 - q^k), {k, 1, n}], z]
and the recursion is:
A[n]=x^n*(x+1/(1+x^n))*A[n-1]
For:
Sum[A[n],{n,0,Infinity]]=1 + Sum[x^(n*(n + 1))/2*Product[x + 1/(1 -x^k),{k, 1, n}], {n, 1, Infinity}].
(End)
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REFERENCES
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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
George E. Andrews, Number Theory,Dover Publications,N.Y. 1971, pp 167-169 [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Oct 21 2009]
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LINKS
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S. Corteel and D. Gouyou-Beauchamps, Enumerations of Sand Piles, Discrete Maths, 256 (2002) 3, 625-643.
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
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FORMULA
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Only complicated recursive formulae are known, see Latapy et al.
G.f.: 1+Sum_{n=1..infinity} x^(n*(n+1)/2)*Product_{k=1..n} (x+1/(1-x^k)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 09 2007
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EXAMPLE
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The fith term of the series is 5 since SPM(5) = { (5), (4,1), (3,2), (3,1,1), (2,2,1) }. The seventh term of the series 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) }.
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MATHEMATICA
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Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Oct 21 2009: (Start)
A[0] = 1; A[n_] := A[n] = x^n*(x + 1/(1 + x^n))*A[n - 1]
Table[A[n], {n, 0, 10}]
(End)
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CROSSREFS
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Sequence in context: A089676 A062436 A121269 this_sequence A085140 A138883 A107849
Adjacent sequences: A056216 A056217 A056218 this_sequence A056220 A056221 A056222
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KEYWORD
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nonn,nice
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AUTHOR
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Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Aug 03 2000
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