A000929 Dimension of n-th degree part of Steenrod algebra.

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 20, 22, 23, 26, 28, 29, 32, 35, 37, 41, 45, 47, 51, 55, 58, 63, 68, 72, 77, 82, 86, 92, 98, 103, 111, 118, 123, 131, 139, 145, 154, 164, 171, 180, 190, 198, 208, 219, 229, 241, 253, 264, 278, 291

Offset: 0

Author

J. Daniel Christensen, Mar 15 1996

Keywords

• nonn

Comments

Number of partitions p(1)+p(2)+...+p(m) = n (into positive parts) such that 2*p(k) <= p(k-1).

Number of partitions of n into parts of the form 2^j-1, j=1,2,... (called s-partitions). Example: a(7)=4 because we have [7], [3,3,1], [3,1,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 06 2006

One direction of a bijection between both sorts of partitions, as an algorithm: take a partition P (p(1)+p(2)+...+p(m) such that 2*p(k) <= p(k-1), m is the number of parts), subtract 1 from p(m), 2 from p(m-1), 4 from p(m-2), etc. (this gives a valid partition of the same type), add the part 2^m-1 to the other (initially empty) partition P', repeat until P is empty. The other direction goes by splitting parts 2^k-1 (uniquely) into distinct powers of 2 that are (in decreasing order) added at the left. - Joerg Arndt, Jan 06 2013

Examples

			From _Joerg Arndt_, Dec 28 2012: (Start)
There are a(17)=13 partitions of 17 into Mersenne numbers:
[ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
[ 2]  [ 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
[ 3]  [ 3 3 1 1 1 1 1 1 1 1 1 1 1 ]
[ 4]  [ 3 3 3 1 1 1 1 1 1 1 1 ]
[ 5]  [ 3 3 3 3 1 1 1 1 1 ]
[ 6]  [ 3 3 3 3 3 1 1 ]
[ 7]  [ 7 1 1 1 1 1 1 1 1 1 1 ]
[ 8]  [ 7 3 1 1 1 1 1 1 1 ]
[ 9]  [ 7 3 3 1 1 1 1 ]
[10]  [ 7 3 3 3 1 ]
[11]  [ 7 7 1 1 1 ]
[12]  [ 7 7 3 ]
[13]  [ 15 1 1 ]
There are a(17)=13 partitions p(1)+p(2)+...+p(m) = 17 such that 2*p(k) <= p(k-1):
[ 1]  [ 10 4 2 1 ]
[ 2]  [ 10 5 2 ]
[ 3]  [ 11 4 2 ]
[ 4]  [ 11 5 1 ]
[ 5]  [ 12 4 1 ]
[ 6]  [ 12 5 ]
[ 7]  [ 13 3 1 ]
[ 8]  [ 13 4 ]
[ 9]  [ 14 2 1 ]
[10]  [ 14 3 ]
[11]  [ 15 2 ]
[12]  [ 16 1 ]
[13]  [ 17 ]
(End)
		

References

• Steenrod, N. and Epstein, D., Cohomology Operations, Princeton Univ. Press, 1962.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 513 terms from Reinhard Zumkeller)
• P. C. P. Bhatt, An interesting way to partition a number, Inform. Process. Lett., 71, 1999, 141-148.
• W. M. Y. Goh, P. Hitczenko and A. Shokoufandeh, s-partitions, Inform. Process. Lett., 82, 2002, 327-329.
• Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.

Crossrefs

Cf. A000225, A000041, A018819, A079559, A117145.

Programs

Maple

The sequence is C(n,n) where C := proc(m,n) option remember; local k, a; if m = 0 then if n = 0 then 1 else 0 fi; elif m > n then C(n,n); else a := 0; for k from 0 to m do a := a + C(floor(k/2), n-k) od; a; fi end;
g:=1/product(1-x^(2^k-1),k=1..10): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..64); # Emeric Deutsch, Mar 06 2006
# alternative Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, min(n-j, iquo(j, 2))), j=1..i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 14 2021

Mathematica

nn = 63; CoefficientList[
Series[Product[1/(1 - x^(2^i - 1)), {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 09 2013 *)

PARI

N=166; q='q+O('q^N);
gf=1/prod(n=1,1+ceil(log(N)/log(2)), 1-q^(2^n - 1) );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

Formula

G.f.: 1/Product_{i>=1} (1 - x^(2^i-1)). - Simon Plouffe (corrected by Joerg Arndt, Dec 28 2012)

a(n) = p(n,1) with p(n,k) = if k <= n then p(n-k,k) + p(n,2*k+1), otherwise 0^n. - Reinhard Zumkeller, Mar 18 2009

G.f.: Sum_{i>=0} x^(2^i-1) / Product_{j=1..i} (1 - x^(2^j-1)). - Ilya Gutkovskiy, Jun 05 2017