A072706 Number of unimodal partitions/compositions of n into distinct terms.

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 33, 39, 55, 69, 93, 127, 159, 201, 261, 327, 411, 537, 653, 819, 1011, 1257, 1529, 1899, 2331, 2829, 3441, 4179, 5031, 6093, 7305, 8767, 10575, 12573, 14997, 17847, 21223, 25089, 29757, 35055, 41379, 48801, 57285, 67131

Offset: 0

Author

Henry Bottomley, Jul 04 2002

Keywords

• nonn

Comments

Also the number of ways to partition a strict integer partition of n into two unordered blocks. - Gus Wiseman, Dec 31 2019

Examples

			a(6)=9 since 6 can be written as 1+2+3, 1+3+2, 1+5, 2+3+1, 2+4, 3+2+1, 4+2, 5+1, or 6, but not for example 1+4+1 (which does not have distinct terms) nor 2+1+3 (which is not unimodal).
From _Joerg Arndt_, Mar 25 2014: (Start)
The a(10) = 33 such compositions of 10 are:
01:  [ 1 2 3 4 ]
02:  [ 1 2 4 3 ]
03:  [ 1 2 7 ]
04:  [ 1 3 4 2 ]
05:  [ 1 3 6 ]
06:  [ 1 4 3 2 ]
07:  [ 1 4 5 ]
08:  [ 1 5 4 ]
09:  [ 1 6 3 ]
10:  [ 1 7 2 ]
11:  [ 1 9 ]
12:  [ 2 3 4 1 ]
13:  [ 2 3 5 ]
14:  [ 2 4 3 1 ]
15:  [ 2 5 3 ]
16:  [ 2 7 1 ]
17:  [ 2 8 ]
18:  [ 3 4 2 1 ]
19:  [ 3 5 2 ]
20:  [ 3 6 1 ]
21:  [ 3 7 ]
22:  [ 4 3 2 1 ]
23:  [ 4 5 1 ]
24:  [ 4 6 ]
25:  [ 5 3 2 ]
26:  [ 5 4 1 ]
27:  [ 6 3 1 ]
28:  [ 6 4 ]
29:  [ 7 2 1 ]
30:  [ 7 3 ]
31:  [ 8 2 ]
32:  [ 9 1 ]
33:  [ 10 ]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

The non-strict version is A001523.

Cf. A000009, A000041, A001970, A032020, A059618, A072705, A072707, A270995, A294617.

Programs

Maple

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 25 2014

Mathematica

b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, Expand[b[n, i - 1] + If[i > n, 0, x*b[n - i, i - 1]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*Ceiling[2^(i - 1)], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
Table[If[n==0,1,Sum[2^(Length[ptn]-1),{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,15}] (* Gus Wiseman, Dec 31 2019 *)

PARI

N=66; q='q+O('q^N); Vec( 1 + sum(n=1, N, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ) ) \\ Joerg Arndt, Mar 25 2014

Formula

a(n) = sum_k A072705(n, k) = A032020(n)-A072707(k) = A032302(n)/2 (n>0).

G.f.: 1/2*(1+Product_{k>0} (1+2*x^k)). - Vladeta Jovovic, Jun 24 2003

G.f.: 1 + sum(n>=1, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Jan 20 2014]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi)*n^(3/4)), where c = -polylog(2, -2) = A266576 = 1.436746366883680946362902023893583354... - Vaclav Kotesovec, Sep 22 2019