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

0,7

Comments

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

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.

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

Example

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

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]

Crossrefs

Lower bound on A076269.

Cf. A161680, A264034.

Sequence in context: A099773 A140471 A029061 * A334576 A081607 A029060

Adjacent sequences: A337203 A337204 A337205 * A337207 A337208 A337209

Keyword

nonn

Author

Grant Kopitzke, Aug 18 2020

Extensions

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

Status

approved