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A195012
Sum of positive cranks minus the sum of positive ranks of all partitions of n.
10
1, 1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 24, 31, 40, 53, 69, 88, 113, 144, 183, 231, 290, 362, 453, 563, 696, 859, 1058, 1296, 1587, 1935, 2354, 2856, 3458, 4175, 5033, 6051, 7259, 8692, 10390, 12391, 14756, 17537, 20808, 24648, 29151, 34417, 40581, 47773, 56158
OFFSET
1,4
COMMENTS
It appears this is also the column 0 of triangle A195011 without the first one (see the Andrews-Garvan-Liang paper, page 16).
Is this also the ospt(n) function mentioned in the Andrews-Chan-Kim paper? Is A115995(n) the first crank momment? Is A209616(n) the first rank moment? - Omar E. Pol, Apr 07 2012
From Jeremy Lovejoy, Oct 14 2022: (Start)
a(n) is also the number of rank 0 strongly unimodal sequences of size n. A strongly unimodal sequence is a sequence of positive integers which are strictly increasing up to a point (the peak) and then strictly decreasing thereafter. The size is the sum of all of the parts and the rank is the number of parts to the left of the peak minus the number of parts to the right of the peak.
For example, there are 10 strongly unimodal sequences of size 6: (6), (1,5), (5,1), (2,4), (4,2), (1,4,1), (3,2,1), (1,2,3), (1,3,2), and (2,3,1). The sequences (6), (1,4,1), (1,3,2), and (2,3,1) have rank 0, and so a(6) = 4. (End)
LINKS
G. E. Andrews, S. H. G. Chan, and B. Kim, The odd moments of ranks and cranks (This is the function C_1 - R_1 of that paper), Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91.
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function, The Ramanujan Journal, December 2012, Volume 29, Issue 1-3, pp 321-338.
A. O. L. Atkin and F. G. Garvan, Relations between the ranks and cranks of partitions, arXiv:math/0208050 [math.NT], 2002.
K. Bringmann, C. Jennings-Shaffer, K. Mahlburg, and R. Rhoades, Peak positions of strongly unimodal sequences, Trans. Amer. Math. Soc. 372 (2019), 7087-7109.
K. Hikami and J. Lovejoy, Torus knots and quantum modular forms, Res. Math. Sci. 2, Article 2 (2015).
FORMULA
a(n) = A115995(n) - A209616(n).
From Jeremy Lovejoy, Oct 14 2022: (Start)
G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n>=1} x^(n*(n+1)/2)*(-1)^(n-1)*(1-x^(n^2))/(1-x^n).
G.f.: (1/Product_{n>=1}(1-x^n))*Sum_{n,r>=0} (-1)^(n+r)*x^(n*(3*n+5)/2+2*n*r+r*(r+3)/2). (End)
EXAMPLE
For n = 6 we have:
------------------------------------------------
Partitions
of 6 Crank Rank
------------------------------------------------
6 6 6 - 1 = 5
3+3 3 3 - 2 = 1
4+2 4 4 - 2 = 2
2+2+2 2 2 - 3 = -1
5+1 1 - 1 = 0 5 - 2 = 3
3+2+1 2 - 1 = 1 3 - 3 = 0
4+1+1 1 - 2 = -1 4 - 3 = 1
2+2+1+1 0 - 2 = -2 2 - 4 = -2
3+1+1+1 0 - 3 = -3 3 - 4 = -1
2+1+1+1+1 0 - 4 = -4 2 - 5 = -3
1+1+1+1+1+1 0 - 6 = -6 1 - 6 = -5
------------------------------------------------
The sum of positive cranks is 6+3+4+2+1 = 16 and the sum of positive ranks is 5+1+2+3+1 = 12 therefore a(6) = 16 - 12 = 4.
MAPLE
# Based on Theorem 1 of Andrews-Chan-Kim:
M:=101;
qinf:=mul(1-q^i, i=1..M);
qinf:=series(qinf, q, M);
C1:=add((-1)^(n+1)*q^(n*(n+1)/2)/(1-q^n), n=1..M);
C1:=series(C1/qinf, q, M);
R1:=add((-1)^(n+1)*q^(n*(3*n+1)/2)/(1-q^n), n=1..M);
R1:=series(R1/qinf, q, M);
series(C1-R1, q, M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012
MATHEMATICA
M = 101;
qinf = Product[1-q^i, {i, 1, M}];
qinf = Series[qinf, {q, 0, M}];
C1 = Sum[(-1)^(n+1) q^(n(n+1)/2)/(1-q^n), {n, 1, M}];
C1 = Series[C1/qinf, {q, 0, M}];
R1 = Sum[(-1)^(n+1) q^(n(3n+1)/2)/(1-q^n), {n, 1, M}];
R1 = Series[R1/qinf, {q, 0, M}];
CoefficientList[Series[C1-R1, {q, 0, M}], q] // Rest (* Jean-François Alcover, Aug 18 2018, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 10 2012
EXTENSIONS
New name, example and more terms from Omar E. Pol, Apr 06 2012
More terms a(44)-a(50) from Alois P. Heinz, Apr 08 2012
STATUS
approved