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A337206 Cardinality of maximal level sets of Gini index on integer partitions.

%I A337206 #28 Jan 20 2023 07:54:20
%S A337206 1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,7,8,9,11,13,15,17,21,23,28,33,38,44,52,
%T A337206 60,72,81,95,112,128,147,175,195,233,267,305,353,412,462,533,617,703,
%U A337206 807,932,1052,1210,1389,1569,1785,2060,2315,2642,3023,3405,3876,4413,4968
%N A337206 Cardinality of maximal level sets of Gini index on integer partitions.
%C A337206 a(n) is a lower bound on A076269(n).
%H A337206 Alois P. Heinz, <a href="/A337206/b337206.txt">Table of n, a(n) for n = 0..300</a>
%H A337206 Grant Kopitzke, <a href="https://arxiv.org/abs/2005.04284">The Gini Index of an Integer Partition</a>, arXiv:2005.04284 [math.CO], 2020.
%F A337206 G.f.: Product_{n=1..oo} 1/(1-q^(binomial(n+1,2))x^n)-1 = Sum_{n=1..oo} Sum_{lambda a partition of n} q^(binomial(n+1,2)-g(lambda))x^n, where g(lambda) is the Gini index of lambda.
%F A337206 a(n) = max_{k=0..A161680(n)} A264034(n,k). - _Alois P. Heinz_, Jan 20 2023
%e A337206 For n=6 the maximal level set of the Gini index contains the partitions (3,3) and (4,1,1). So a(6)=2.
%p A337206 b:= proc(n, i, w) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A337206       b(n, i-1, w)+expand(x^(w*i)*b(n-i, min(i, n-i), w+1))))
%p A337206     end:
%p A337206 a:= n-> max(coeffs(b(n$2, 0))):
%p A337206 seq(a(n), n=0..61);  # _Alois P. Heinz_, Jan 20 2023
%t A337206 m = 75;
%t A337206 p = Product[ 1/(1 - q^Binomial[i + 1, 2] x^i), {i, 1, m}];
%t A337206 psn = Expand@Normal@Series[ p, {x, 0, m}];
%t A337206 psnc = CoefficientList[CoefficientList[psn, {x}, {m}], {q}];
%t A337206 Map[Max, psnc]
%Y A337206 Lower bound on A076269.
%Y A337206 Cf. A161680, A264034.
%K A337206 nonn
%O A337206 0,7
%A A337206 _Grant Kopitzke_, Aug 18 2020
%E A337206 Typo in a(43) corrected by _Alois P. Heinz_, Jan 20 2023