A064428 - OEIS

A064428

Number of partitions of n with nonnegative crank.

1, 0, 1, 2, 3, 4, 6, 8, 12, 16, 23, 30, 42, 54, 73, 94, 124, 158, 206, 260, 334, 420, 532, 664, 835, 1034, 1288, 1588, 1962, 2404, 2953, 3598, 4392, 5328, 6466, 7808, 9432, 11338, 13632, 16326, 19544, 23316, 27806, 33054, 39273, 46534, 55096, 65076, 76808

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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 permutations is A000166, complement A002467.

The version for compositions is A238351.

This is column k = 0 of A352833.

A238352 counts reversed partitions by fixed points, rank statistic A352822.

A238394 counts reversed partitions without a fixed point, ranked by A352830.

A238395 counts reversed partitions with a fixed point, ranked by A352872. (End)

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

a(n) = (A000041(n) + A064410(n)) / 2, n>1. - Michael Somos, Jul 28 2003

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

a(n) = A000041(n) - A001522(n). - Gus Wiseman, Mar 30 2021

a(n) = A064410(n) + A001522(n). - Gus Wiseman, May 21 2022

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

These are the row-sums of the right (or left) half of A064391, inclusive.

The case of crank 0 is A064410, ranked by A342192.

The strict case is A352828.

These partitions are ranked by A352873.

A000700 = self-conjugate partitions, ranked by A088902, complement A330644.

A001522 counts partitions with positive crank, ranked by A352874.

A034008 counts even-length compositions.

A115720 and A115994 count partitions by their Durfee square.

A224958 counts compositions w/ alternating parts unequal (even: A342532).

A257989 gives the crank of the partition with Heinz number n.

A342527 counts compositions w/ alternating parts equal (even: A065608).