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A338085
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a(n) is the cardinality of S(n), the subset of partitions of n such that there are enough smaller parts to add together to be greater than a larger part.
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0
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0, 0, 0, 0, 1, 1, 5, 5, 12, 18, 30, 36, 65, 83, 120, 159, 225, 284, 395, 495, 665, 848, 1094, 1348, 1757, 2184, 2746, 3399, 4250, 5199, 6469, 7867, 9667, 11756, 14310, 17266, 20988, 25216, 30372, 36371, 43648, 52041, 62187, 73866, 87837, 104105, 123279, 145453
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OFFSET
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1,7
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COMMENTS
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In George Andrews’s partition notation, exponents mean repeated addition, not repeated multiplication. So (p^K)(q^L) with p<q denotes K parts of size p, combined with L parts of size q, thus a rectangle of width q and depth L atop another rectangle of width p and depth L in the Ferrers diagram. Given a partition Lambda(n)=(p1^E1)(p2^E2)...(pj^Ej) with all Ek>0 and the parts pk arranged in increasing order, suppose E1p1+E2p2+..Ekpk>p(k+1) for some 1<k<j. Then Lambda(n) is in S(n).
For any partition in S, the number of parts must be at least 3, with at least 2 distinct parts.
The sequence a(n) is nondecreasing since if a(n-1)=t, then t distinct elements of S(n) can be formed by putting a dot in the lower left corner of the Ferrers diagram for each element of S(n-1).
Closure: Given a partition X in S(x) and partition Y in S(y): The partition X+Y given by concatenation is in S(x+y). So a(x)+a(y) might be provably less than or equal to S(x+y). X and Y can be multiplied to give a partition XY in S(xy). The two operations obey distributivity of multiplication over addition.
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LINKS
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EXAMPLE
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(4^2)(7^3), a partition of 29, is in S(29) since 2*4=8>7.
Also, (1^3)(3^2)(7^1)(20^4), a partition of 96, is in S(96) since 3*1+2*3=9>7.
But (1^3)(4^5) is not in S(23) because 3*1 is not greater than 4.
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MATHEMATICA
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ispart[p_] := Module[{s = 0}, For[i = 1, i <= Length[p], i++, If[s > p[[i]] && p[[i]] > p[[i-1]], Return[1]]; s += p[[i]]]; 0];
a[n_] := a[n] = Module[{c = 0}, Do[ c += ispart[p], {p, Reverse /@ IntegerPartitions[n]}]; c];
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PROG
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(PARI)
ispart(p)={my(s=0); for(i=1, #p, if(s>p[i]&&p[i]>p[i-1], return(1)); s+=p[i]); 0}
a(n)={my(c=0); forpart(p=n, c+=ispart(p)); c} \\ Andrew Howroyd, Oct 25 2020
(PARI)
a(n)={local(Cache=Map()); my(F(r, k, b)=my(hk=[r, k, b], z); if(!mapisdefined(Cache, hk, &z), z = if(k<=1, b, sum(m=0, r\k, self()(r-m*k, k-1, b||(m&&r-m*k>k)))); mapput(Cache, hk, z)); z); F(n, n, 0)} \\ Andrew Howroyd, Nov 03 2020
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CROSSREFS
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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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