OFFSET
0,3
COMMENTS
Also number of partitions of n such that all parts, with the possible exception of the largest, appear only once. Example: a(6)=7 because we have [6], [5,1], [4,2], [3,3], [3,2,1], [2,2,2] and [1,1,1,1,1,1] ([4,1,1], [3,1,1,1], [2,2,1,1], [2,1,1,1,1,1] do not qualify). - Emeric Deutsch and Vladeta Jovovic, Feb 23 2006
Also the number of partitions p of n such that d(p) > max(p) - min(p), where d is the number of distinct parts of p; indeed that inequality occurs only when d(p) = 1+ max(p) - min(p), so p satisfies a(i) - a(i-1) = 1 for all parts, ordered as a(i) >= a(i-1) > ... > a(k). - Clark Kimberling, Apr 18 2014
Flat partitions are also called gap-free partitions. See, for example, the Grabner et al. reference. - Emeric Deutsch, Sep 22 2016
Conjecture: Also the sum of the smallest parts in the distinct partitions of n with an odd number of parts. - George Beck, May 06 2017
Above conjecture was proved by Shane Chern, see link. - George Beck, Aug 12 2017
Note that Andrews [2016] uses a(0) = 1. - Michael Somos, Aug 07 2017
Also called number of compact partitions of n where a compact partition is one where every integer between the largest and smallest parts of it also appears as a part. [Andrews 2016] - Michael Somos, Aug 13 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016; See page 4 equation (2.1).
T. M. Brown, Convex Domino Towers, J. Int. Seq. 20 (2017) # 17.3.1
Shane Chern, On a conjecture of George Beck, arXiv:1705.10700 [math.NT], 2017.
P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018. Also Discrete Masth., 343 (2020), # 111875.
Jane Y. X. Yang, Combinatorial proofs and generalizations on conjectures related with Euler's partition theorem, arXiv:1801.06815 [math.CO], 2018.
FORMULA
G.f.: x/(1-x) + x^2/(1-x^2)*(1+x) + x^3/(1-x^3)*(1+x)*(1+x^2) + x^4/(1-x^4)*(1+x)*(1+x^2)*(1+x^3) + x^5/(1-x^5)*(1+x)*(1+x^2)*(1+x^3)*(1+x^4) + ... . - Emeric Deutsch and Vladeta Jovovic, Feb 22 2006
a(n) = Sum_{k=0..1} A238353(n,k). - Alois P. Heinz, Mar 09 2014
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 24 2018
EXAMPLE
From Joerg Arndt, Dec 27 2012: (Start)
The a(11)=18 flat partitions of 11 are (in lexicographic order)
[ 1] [ 1 1 1 1 1 1 1 1 1 1 1 ]
[ 2] [ 2 1 1 1 1 1 1 1 1 1 ]
[ 3] [ 2 2 1 1 1 1 1 1 1 ]
[ 4] [ 2 2 2 1 1 1 1 1 ]
[ 5] [ 2 2 2 2 1 1 1 ]
[ 6] [ 2 2 2 2 2 1 ]
[ 7] [ 3 2 1 1 1 1 1 1 ]
[ 8] [ 3 2 2 1 1 1 1 ]
[ 9] [ 3 2 2 2 1 1 ]
[10] [ 3 2 2 2 2 ]
[11] [ 3 3 2 1 1 1 ]
[12] [ 3 3 2 2 1 ]
[13] [ 3 3 3 2 ]
[14] [ 4 3 2 1 1 ]
[15] [ 4 3 2 2 ]
[16] [ 4 4 3 ]
[17] [ 6 5 ]
[18] [ 11 ]
The a(11)=18 partitions of 11 where no part (except possibly the largest) is repeated are
[ 1] [ 1 1 1 1 1 1 1 1 1 1 1 ]
[ 2] [ 2 2 2 2 2 1 ]
[ 3] [ 3 3 3 2 ]
[ 4] [ 4 4 2 1 ]
[ 5] [ 4 4 3 ]
[ 6] [ 5 3 2 1 ]
[ 7] [ 5 4 2 ]
[ 8] [ 5 5 1 ]
[ 9] [ 6 3 2 ]
[10] [ 6 4 1 ]
[11] [ 6 5 ]
[12] [ 7 3 1 ]
[13] [ 7 4 ]
[14] [ 8 2 1 ]
[15] [ 8 3 ]
[16] [ 9 2 ]
[17] [ 10 1 ]
[18] [ 11 ]
(End)
MAPLE
g:= 1+sum(x^j*product(1+x^i, i=1..j-1)/(1-x^j), j=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..50); # Emeric Deutsch, Feb 23 2006
# Alternative:
b:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1), j=1..n/i)))
end:
a:= n-> add(b(n, k), k=0..n):
seq(a(n), n=0..70); # Alois P. Heinz, Jul 06 2012
MATHEMATICA
nn=54; Drop[CoefficientList[Series[Sum[x^i/(1-x^i)Product[1+x^j, {j, 1, i-1}], {i, 1, nn}], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Sep 28 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[b[n, k], {k, 1, n}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ Sum[ x^k / (1 - x^k) QPochhammer[ -x, x, k - 1] // FunctionExpand, {k, n}], {x, 0, n}]; (* Michael Somos, Aug 07 2017 *)
PROG
(PARI)
N = 66; x = 'x + O('x^N);
gf = sum(n=1, N, x^n/(1-x^n) * prod(k=1, n-1, 1+x^k) );
v = Vec(gf)
/* Joerg Arndt, Apr 21 2013 */
(PARI) {a(n) = my(t); if( n<1, 0, polcoeff(sum(k=1, n, (t *= 1 + x^k) * x^k / (1 - x^(2*k)), t = 1 + x * O(x^n)), n))}; /* Michael Somos, Aug 07 2017 */
(PARI) {a(n) = my(c); forpart(p=n, c++; for(i=1, #p-1, if( p[i+1] > p[i] + 1, c--; break))); c}; /* Michael Somos, Aug 13 2017 */
(Python)
from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==0 else 0 if i<1 else sum(b(n - i*j, i - 1) for j in range(1, n//i + 1))
def a(n): return sum(b(n, k) for k in range(n + 1))
print([a(n) for n in range(71)]) # Indranil Ghosh, Aug 14 2017, after Maple code by Alois P. Heinz
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Feb 23 2006
a(0)=1 prepended by Alois P. Heinz, Aug 14 2017
STATUS
approved