A211859 Number of partitions of n into parts <= 4 with the property that all parts have distinct multiplicities.
1, 1, 2, 2, 4, 4, 6, 8, 10, 10, 14, 18, 18, 26, 31, 30, 39, 48, 48, 61, 63, 73, 84, 101, 98, 124, 132, 147, 156, 188, 182, 223, 227, 257, 272, 322, 306, 367, 377, 417, 427, 499, 488, 564, 567, 645, 647, 740, 720, 828, 836, 920, 924, 1048, 1030, 1173, 1161
Offset: 0
Examples
For n=3 the a(3) = 2 partitions are [3] and [1,1,1]. Note that [2,1] does not count, as 1 and 2 appear with the same nonzero multiplicity.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.
Programs
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Haskell
a211859 n = p 0 [] [1..4] n where p m ms _ 0 = if m `elem` ms then 0 else 1 p [] _ = 0 p m ms ks'@(k:ks) x | x < k = 0 | m == 0 = p 1 ms ks' (x - k) + p 0 ms ks x | m `elem` ms = p (m + 1) ms ks' (x - k) | otherwise = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x -- Reinhard Zumkeller, Dec 27 2012
Formula
G.f.: (9*x^45 +20*x^44 +44*x^43 +76*x^42 +121*x^41 +172*x^40 +234*x^39 +292*x^38 +346*x^37 +380*x^36 +412*x^35 +415*x^34 +417*x^33 +401*x^32 +389*x^31 +365*x^30 +361*x^29 +351*x^28 +359*x^27 +365*x^26 +383*x^25 +391*x^24 +413*x^23 +422*x^22 +436*x^21 +444*x^20 +454*x^19 +454*x^18 +458*x^17 +450*x^16 +437*x^15 +415*x^14 +383*x^13 +342*x^12 +298*x^11 +248*x^10 +198*x^9 +152*x^8 +110*x^7 +76*x^6 +49*x^5 +30*x^4 +16*x^3 +8*x^2 +3*x +1) / ((x^2-x+1) *(x^4-x^3+x^2-x+1) *(x^6+x^3+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+x+1)^2 *(x^4+x^3+x^2+x+1)^2 *(x^2+1)^2 *(x+1)^3 *(x-1)^4). - Alois P. Heinz, Feb 09 2017