OFFSET
0,4
COMMENTS
For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
From Gus Wiseman, Mar 30 2021 and May 21 2022: (Start)
Also the number of even-length compositions of n with alternating parts strictly decreasing, or properly 2-colored partitions (proper = no equal parts of the same color) with the same number of parts of each color, or ordered pairs of strict partitions of the same length with total n. The odd-length case is A001522, and there are a total of A000041 compositions with alternating parts strictly decreasing (see A342528 for a bijective proof). The a(2) = 1 through a(7) = 8 ordered pairs of strict partitions of the same length are:
(1)(1) (1)(2) (1)(3) (1)(4) (1)(5) (1)(6)
(2)(1) (2)(2) (2)(3) (2)(4) (2)(5)
(3)(1) (3)(2) (3)(3) (3)(4)
(4)(1) (4)(2) (4)(3)
(5)(1) (5)(2)
(21)(21) (6)(1)
(21)(31)
(31)(21)
Conjecture: Also the number of integer partitions y of n without a fixed point y(i) = i, ranked by A352826. This is stated at A238394, but Resta tells me he may not have had a proof. The a(2) = 1 through a(7) = 8 partitions without a fixed point are:
(2) (3) (4) (5) (6) (7)
(21) (31) (41) (33) (43)
(211) (311) (51) (61)
(2111) (411) (331)
(3111) (511)
(21111) (4111)
(31111)
(211111)
The version for compositions is A238351.
This is column k = 0 of A352833.
The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - Jeremy Lovejoy, Sep 26 2022
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (i).
G. E. Andrews, B. C. Berndt, Ramanujan's Lost Notebook Part I, Springer, see p. 169 Entry 6.7.1.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Cody Armond and Oliver T. Dasbach, Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial, arXiv:1106.3948 [math.GT], 2011.
Cristina Ballantine and Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Rupam Barman and Ajit Singh, On Mex-related partition functions of Andrews and Newman, arXiv:2009.11602 [math.NT], 2020.
Aubrey Blecher and Arnold Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.
Brian Hopkins, James A. Sellers, and Ae Ja Yee, Combinatorial Perspectives on the Crank and Mex Partition Statistics, arXiv:2108.09414 [math.CO], 2021.
Mbavhalelo Mulokwe and Konstantinos Zoubos, Free fermions, neutrality and modular transformations, arXiv:2403.08531 [hep-th], 2024.
FORMULA
G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1-x^k). - Michael Somos, Jul 28 2003
G.f.: Sum_{i>=0} x^(i*(i+1)) / (Product_{j=1..i} 1-x^j )^2. - Jon Perry, Jul 18 2004
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Sep 26 2016
G.f.: (Sum_{i>=0} x^i / (Product_{j=1..i} 1-x^j)^2 ) * (Product_{k>0} 1-x^k). - Li Han, May 23 2020
EXAMPLE
G.f. = 1 + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + 23*x^10 + ... - Michael Somos, Jan 15 2018
From Gus Wiseman, May 21 2022: (Start)
The a(0) = 1 through a(8) = 12 partitions with nonnegative crank:
() . (2) (3) (4) (5) (6) (7) (8)
(21) (22) (32) (33) (43) (44)
(31) (41) (42) (52) (53)
(221) (51) (61) (62)
(222) (322) (71)
(321) (331) (332)
(421) (422)
(2221) (431)
(521)
(2222)
(3221)
(3311)
(End)
MATHEMATICA
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^(k (k + 1)/2) , {k, 0, (Sqrt[1 + 8 n] - 1)/2}] / QPochhammer[ x], {x, 0, n}]]; (* Michael Somos, Jan 15 2018 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k (k + 1)) / QPochhammer[ x, x, k]^2 , {k, 0, (Sqrt[1 + 4 n] - 1)/2}], {x, 0, n}]]; (* Michael Somos, Jan 15 2018 *)
ck[y_]:=With[{w=Count[y, 1]}, If[w==0, If[y=={}, 0, Max@@y], Count[y, _?(#>w&)]-w]]; Table[Length[Select[IntegerPartitions[n], ck[#]>=0&]], {n, 0, 30}] (* Gus Wiseman, Mar 30 2021 *)
ici[q_]:=And@@Table[q[[i]]>q[[i+2]], {i, Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], EvenQ@*Length], ici]], {n, 0, 15}] (* Gus Wiseman, Mar 30 2021 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) -1)\2, (-1)^k * x^((k+k^2)/2)) / eta( x + x * O(x^n)), n))}; /* Michael Somos, Jul 28 2003 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Sep 30 2001
STATUS
approved