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A064428 Number of partitions of n with nonnegative crank. 15
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 (list; graph; refs; listen; history; text; internal format)
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: (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). For example, the a(2) = 1 through a(9) = 16 ordered pairs of strict partitions of the same length are:

  (1)(1)  (1)(2)  (1)(3)  (1)(4)   (1)(5)    (1)(6)    (1)(7)    (1)(8)

          (2)(1)  (2)(2)  (2)(3)   (2)(4)    (2)(5)    (2)(6)    (2)(7)

                  (3)(1)  (3)(2)   (3)(3)    (3)(4)    (3)(5)    (3)(6)

                          (4)(1)   (4)(2)    (4)(3)    (4)(4)    (4)(5)

                                   (5)(1)    (5)(2)    (5)(3)    (5)(4)

                                  (21)(21)   (6)(1)    (6)(2)    (6)(3)

                                            (21)(31)   (7)(1)    (7)(2)

                                            (31)(21)  (21)(32)   (8)(1)

                                                      (21)(41)  (21)(42)

                                                      (31)(31)  (21)(51)

                                                      (32)(21)  (31)(32)

                                                      (41)(21)  (31)(41)

                                                                (32)(31)

                                                                (41)(31)

                                                                (42)(21)

                                                                (51)(21)

(End)

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.

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

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

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

A001522 counts partitions with positive crank.

A034008 counts even-length compositions.

A064391 counts partitions by crank.

A064410 counts partitions of crank 0.

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).

A342528 = compositions w/ alternating parts weakly decr. (even: A114921).

Cf. A000041, A000726, A003242, A008965, A062968, A325547, A325548.

Sequence in context: A241828 A125895 A241344 * A052810 A320315 A164090

Adjacent sequences:  A064425 A064426 A064427 * A064429 A064430 A064431

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Sep 30 2001

STATUS

approved

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Last modified July 25 13:05 EDT 2021. Contains 346290 sequences. (Running on oeis4.)