A064391 Triangle T(n,k) with zeroth row {1} and row n for n >= 1 giving number of partitions of n with crank k, for -n <= k <= n.

1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2

Offset: 0

Author

N. J. A. Sloane, Sep 29 2001

Keywords

• nonn
• tabf
• nice
• easy

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).

n-th row contains 2n+1 terms.

Examples

			{T(20, k), -20 <= k <=20} = {1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 19, 20, 26, 28, 34, 34, 39, 38, 41, 38, 39, 34, 34, 28, 26, 20, 19, 13, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1}.
From _Omar E. Pol_, Mar 04 2012: (Start)
Triangle begins:
.                          1;
.                       1, 0, 0;
.                    1, 0, 0, 0, 1;
.                 1, 0, 0, 1, 0, 0, 1;
.              1, 0, 1, 0, 1, 0, 1, 0, 1;
.           1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1;
.        1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1;
.     1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1;
.  1, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1;
1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 3, 2, 3, 1, 2, 1, 1, 0, 1;
(End)
		

Links

• G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
• F. Garvan, New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc., 305 (1988), 47-77.

Crossrefs

Cf. A001522, A064410, A064428.

Row sums give A000041. - Omar E. Pol, Mar 04 2012

Programs

Mathematica

max = 12; f[k_ /; k < 0] := f[-k]; f[k_] := Sum[(-1)^m*x^(k*m)*(x^((m^2 + m)/2) - x^((m^2 - m)/2)), {m, 1, max}]/Product[1 - x^m, {m, 1, max}]; t = Table[ Series[f[k], {x, 0, max}] // CoefficientList[#, x]&, {k, -(max-2), max-2}] // Transpose; Table[If[n == 2, {1, 0, 0}, Table[t[[n, k]], {k, max-n, max+n-2}]], {n, 1, max-1}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Vladeta Jovovic *)

Sage

for n in (0..9): # computes the sequence as a triangle
    a = [p.crank() for p in Partitions(n)]
    [a.count(k) for k in (-n..n)] # Peter Luschny, Sep 15 2014

Formula

G.f. for k-th column is Sum(m>=1, (-1)^m*x^(k*m)*(x^((m^2+m)/2)-x^((m^2-m)/2)))/Product(m>=1, 1-x^m). - Vladeta Jovovic, Dec 22 2004

Extensions

More terms from Vladeta Jovovic, Sep 29 2001