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A342528
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Number of compositions with alternating parts weakly decreasing (or weakly increasing).
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22
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1, 1, 2, 4, 7, 12, 20, 32, 51, 79, 121, 182, 272, 399, 582, 839, 1200, 1700, 2394, 3342, 4640, 6397, 8771, 11955, 16217, 21878, 29386, 39285, 52301, 69334, 91570, 120465, 157929, 206313, 268644, 348674, 451185, 582074, 748830, 960676, 1229208, 1568716, 1997064
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OFFSET
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0,3
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COMMENTS
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These are finite sequences q of positive integers summing to n such that q(i) >= q(i+2) for all possible i.
The strict case (alternating parts are strictly decreasing) is A000041. Is there a bijective proof?
Yes. Construct a Ferrers diagram by placing odd parts horizontally and even parts vertically in a fishbone pattern. The resulting Ferrers diagram will be for an ordinary partition and the process is reversible. It does not appear that this method can be applied to give a formula for this sequence. - Andrew Howroyd, Mar 25 2021
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LINKS
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EXAMPLE
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The a(1) = 1 through a(6) = 20 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(111) (31) (32) (33)
(121) (41) (42)
(211) (131) (51)
(1111) (212) (141)
(221) (222)
(311) (231)
(1211) (312)
(2111) (321)
(11111) (411)
(1212)
(1311)
(2121)
(2211)
(3111)
(12111)
(21111)
(111111)
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], GreaterEqual@@Plus@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 15}]
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PROG
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(PARI) seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(1+sum(k=1, n, polcoef(p, k, y)*(polcoef(p, k-1, y) + polcoef(p, k, y))))} \\ Andrew Howroyd, Mar 24 2021
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CROSSREFS
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The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts equal is A342527.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
Cf. A001522, A008965, A048004, A059966, A062968, A064410, A064428, A065608, A167606, A325557, A342519.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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