OFFSET
0,5
COMMENTS
This sequence is part of the contribution to the b^2 term of C_{1-b,2}(q) for(1-b,2)-colored partitions - partitions in which we can label parts any of an indeterminate 1-b colors, but are restricted to using only 2 of the colors per part size. This formula is known to match the Han/Nekrasov-Okounkov hooklength formula truncated at hooks of size two up to the linear term in b.
It is of interest to enumerate and determine specific characteristics of partitions of n, considering each partition individually.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398 [math.CO], 2008.
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'institut Fourier, Tome 60 (2010) no. 1, pp. 1-29.
W. J. Keith, Restricted k-color partitions, Ramanujan Journal (2016) 40: 71.
FORMULA
a(n) = Sum_{p in P(n)} (H(2,p)^2 + 2*A024786 - 2*A024788), where P(n) is the set of partitions of n, and H(2,p) is the hooks of length 2 in partition p.
G.f: (q^2*(1+q^4))/((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j).
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - Vaclav Kotesovec, May 22 2018
EXAMPLE
For a(6), we sum over partitions of six. For each partition, we count 1 for each part which appears more than once, then square the total in each partition.
6............0^2 = 0
5,1..........0^2 = 0
4,2..........0^2 = 0
4,1,1........1^2 = 1
3,3..........1^2 = 1
3,2,1........0^2 = 0
3,1,1,1......1^2 = 1
2,2,2........1^2 = 1
2,2,1,1......2^2 = 4
2,1,1,1,1....1^2 = 1
1,1,1,1,1,1..1^2 = 1
--------------------
Total.............10
MAPLE
b:= proc(n, i, p) option remember; `if`(n=0 or i=1, (
`if`(n>1, 1, 0)+p)^2, add(b(n-i*j, i-1,
`if`(j>1, 1, 0)+p), j=0..n/i))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Apr 05 2018
MATHEMATICA
Array[Total[Count[Split@ #, _?(Length@ # > 1 &)]^2 & /@ IntegerPartitions[#]] &, 44] (* Michael De Vlieger, Apr 07 2018 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (
If[n > 1, 1, 0] + p)^2, Sum[b[n - i*j, i - 1,
If[j > 1, 1, 0] + p], {j, 0, n/i}]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)
PROG
(Python)
def sum_square_freqs_greater_one(freq_list):
tot = 0
for f in freq_list:
count = 0
for i in f:
if i > 1:
count += 1
tot += count*count
return tot
def frequencies(partition, n):
tot = 0
freq_list = []
i = 0
for p in partition:
freq = [0 for i in range(n+1)]
for i in p:
freq[i] += 1
for f in freq:
if f == 0:
tot += 1
freq_list.append(freq)
return freq_list
CROSSREFS
KEYWORD
nonn
AUTHOR
Emily Anible, Apr 05 2018
STATUS
approved