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

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, 60, 72, 81, 95, 112, 128, 147, 175, 195, 233, 267, 305, 353, 412, 462, 533, 617, 703, 807, 932, 1052, 1210, 1389, 1569, 1785, 2060, 2315, 2642, 3023, 3405, 3876, 4413, 4968

Offset: 0

Author

Grant Kopitzke, Aug 18 2020

Keywords

• nonn

Comments

a(n) is a lower bound on A076269(n).

Examples

			For n=6 the maximal level set of the Gini index contains the partitions (3,3) and (4,1,1). So a(6)=2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Grant Kopitzke, The Gini Index of an Integer Partition, arXiv:2005.04284 [math.CO], 2020.

Crossrefs

Lower bound on A076269.

Cf. A161680, A264034.

Programs

Maple

b:= proc(n, i, w) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, w)+expand(x^(w*i)*b(n-i, min(i, n-i), w+1))))
    end:
a:= n-> max(coeffs(b(n$2, 0))):
seq(a(n), n=0..61);  # Alois P. Heinz, Jan 20 2023

Mathematica

m = 75;
p = Product[ 1/(1 - q^Binomial[i + 1, 2] x^i), {i, 1, m}];
psn = Expand@Normal@Series[ p, {x, 0, m}];
psnc = CoefficientList[CoefficientList[psn, {x}, {m}], {q}];
Map[Max, psnc]

Formula

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.

a(n) = max_{k=0..A161680(n)} A264034(n,k). - Alois P. Heinz, Jan 20 2023

Extensions

Typo in a(43) corrected by Alois P. Heinz, Jan 20 2023