OFFSET
0,5
COMMENTS
Also number of partitions of n with positive crank (n>=2), cf. A064391. - Vladeta Jovovic, Sep 30 2001
Number of smooth weakly unimodal compositions of n into positive parts such that the first and last part are 1 (smooth means that successive parts differ by at most one), see example. Dropping the requirement for unimodality gives A186085. - Joerg Arndt, Dec 09 2012
Number of weakly unimodal compositions of n where the maximal part m appears at least m times, see example. - Joerg Arndt, Jun 11 2013
Also weakly unimodal compositions of n with first part 1, maximal up-step 1, and no consecutive up-steps; see example. The smooth weakly unimodal compositions are recovered by shifting all rows above the bottom row to the left by one position with respect to the next lower row. - Joerg Arndt, Mar 30 2014
It would seem from Stanley that he regards a(0)=0 for this sequence and A001523. - Michael Somos, Feb 22 2015
From Gus Wiseman, Mar 30 2021: (Start)
Also the number of odd-length compositions of n with alternating parts strictly decreasing. These are finite odd-length sequences q of positive integers summing to n such that q(i) > q(i+2) for all possible i. The even-length version is A064428. For example, the a(1) = 1 through a(9) = 14 compositions are:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(211) (221) (231) (241) (251) (261)
(311) (312) (322) (332) (342)
(321) (331) (341) (351)
(411) (412) (413) (423)
(421) (422) (432)
(511) (431) (441)
(512) (513)
(521) (522)
(611) (531)
(612)
(621)
(711)
(32211)
(End)
In the Ferrers diagram of a partition x of n, count the dots in each diagonal parallel to the main diagonal (starting at the top-right, say). The result diag(x) is a smooth weakly unimodal composition of n into positive parts such that the first and last part are 1. For example, diag(5541) = 11233221. The function diag is many-to-one; the size of its codomain as a set is a(n). If diag(x) = diag(y), each hook of x can be slid by the same amount past the main diagonal to get y. For example, diag(5541) = diag(44331). - George Beck, Sep 26 2021
From Gus Wiseman, May 23 2022: (Start)
Conjecture: Also the number of integer partitions y of n with a fixed point y(i) = i. These partitions are ranked by A352827. The conjecture is stated at A238395, but Resta tells me he may not have had a proof. The a(1) = 1 through a(8) = 10 partitions are:
(1) (11) (111) (22) (32) (42) (52) (62)
(1111) (221) (222) (322) (422)
(11111) (321) (421) (521)
(2211) (2221) (2222)
(111111) (3211) (3221)
(22111) (4211)
(1111111) (22211)
(32111)
(221111)
(11111111)
Note that these are not the same partitions (compare A352827 to A352874), only the same count (apparently).
(End)
The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - Jeremy Lovejoy, Sep 26 2022
REFERENCES
G. E. Andrews, The reasonable and unreasonable effectiveness of number theory in statistical mechanics, pp. 21-34 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Erich Friedman, Illustration of initial terms
A. D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011.
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.
FORMULA
G.f.: 1 + ( Sum_{k>=1} -(-1)^k * x^(k*(k+1)/2) ) / ( Product_{k>=1} 1-x^k ).
G.f.: 1 + ( Sum_{n>=1} q^(n^2) / ( Product_{k=1..n-1} 1-q^k )^2 * (1-q^n) ) ). - Joerg Arndt, Dec 09 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) [Auluck, 1951]. - Vaclav Kotesovec, Sep 26 2016
EXAMPLE
For a(6)=5 we have the following stacks:
.x... ..x.. ...x. .xx.
xxxxx xxxxx xxxxx xxxx xxxxxx
.
From Joerg Arndt, Dec 09 2012: (Start)
There are a(9) = 14 smooth weakly unimodal compositions of 9:
01: [ 1 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 2 1 ]
03: [ 1 1 1 1 1 2 1 1 ]
04: [ 1 1 1 1 2 1 1 1 ]
05: [ 1 1 1 1 2 2 1 ]
06: [ 1 1 1 2 1 1 1 1 ]
07: [ 1 1 1 2 2 1 1 ]
08: [ 1 1 2 1 1 1 1 1 ]
09: [ 1 1 2 2 1 1 1 ]
10: [ 1 1 2 2 2 1 ]
11: [ 1 2 1 1 1 1 1 1 ]
12: [ 1 2 2 1 1 1 1 ]
13: [ 1 2 2 2 1 1 ]
14: [ 1 2 3 2 1 ]
(End)
From Joerg Arndt, Jun 11 2013: (Start)
There are a(9) = 14 weakly unimodal compositions of 9 where the maximal part m appears at least m times:
01: [ 1 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 2 2 ]
03: [ 1 1 1 1 2 2 1 ]
04: [ 1 1 1 2 2 1 1 ]
05: [ 1 1 1 2 2 2 ]
06: [ 1 1 2 2 1 1 1 ]
07: [ 1 1 2 2 2 1 ]
08: [ 1 2 2 1 1 1 1 ]
09: [ 1 2 2 2 1 1 ]
10: [ 1 2 2 2 2 ]
11: [ 2 2 1 1 1 1 1 ]
12: [ 2 2 2 1 1 1 ]
13: [ 2 2 2 2 1 ]
14: [ 3 3 3 ]
(End)
From Joerg Arndt, Mar 30 2014: (Start)
There are a(9) = 14 compositions of 9 with first part 1, maximal up-step 1, and no consecutive up-steps:
01: [ 1 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 1 2 ]
03: [ 1 1 1 1 1 1 2 1 ]
04: [ 1 1 1 1 1 2 1 1 ]
05: [ 1 1 1 1 1 2 2 ]
06: [ 1 1 1 1 2 1 1 1 ]
07: [ 1 1 1 1 2 2 1 ]
08: [ 1 1 1 2 1 1 1 1 ]
09: [ 1 1 1 2 2 1 1 ]
10: [ 1 1 1 2 2 2 ]
11: [ 1 1 2 1 1 1 1 1 ]
12: [ 1 1 2 2 1 1 1 ]
13: [ 1 1 2 2 2 1 ]
14: [ 1 1 2 2 3 ]
(End)
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ...
MAPLE
b:= proc(n, i, t) option remember; `if`(n<=0, `if`(i=1, 1, 0),
`if`(n<0 or i<1, 0, b(n-i, i, t)+b(n-(i-1), i-1, false)+
`if`(t, b(n-(i+1), i+1, t), 0)))
end:
a:= n-> b(n-1, 1, true):
seq(a(n), n=0..70); # Alois P. Heinz, Feb 26 2014
# second Maple program:
A001522 := proc(n)
local r, a;
a := 0 ;
if n = 0 then
return 1 ;
end if;
for r from 1 do
if r*(r+1) > 2*n then
return a;
else
a := a-(-1)^r*combinat[numbpart](n-r*(r+1)/2) ;
end if;
end do:
end proc: # R. J. Mathar, Mar 07 2015
MATHEMATICA
max = 50; f[x_] := 1 + Sum[-(-1)^k*x^(k*(k+1)/2), {k, 1, max}] / Product[(1-x^k), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 27 2011, after g.f. *)
b[n_, i_, t_] := b[n, i, t] = If[n <= 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, b[n-i, i, t] + b[n - (i-1), i-1, False] + If[t, b[n - (i+1), i+1, t], 0]]]; a[n_] := b[n-1, 1, True]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[Sum[(-1)^(j-1)*PartitionsP[n-j*((j+1)/2)], {j, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}], {n, 1, 60}]}] (* Vaclav Kotesovec, Sep 26 2016 *)
ici[q_]:=And@@Table[q[[i]]>q[[i+2]], {i, Length[q]-2}];
Table[If[n==0, 1, Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], OddQ@*Length], ici]]], {n, 0, 15}] (* Gus Wiseman, Mar 30 2021 *)
PROG
(PARI) {a(n) = if( n<1, n==0, polcoeff( sum(k=1, (sqrt(1+8*n) - 1)\2, -(-1)^k * x^((k + k^2)/2)) / eta(x + x * O(x^n)), n))}; /* Michael Somos, Jul 22 2003 */
(PARI) N=66; q='q+O('q^N);
Vec( 1 + sum(n=1, N, q^(n^2)/(prod(k=1, n-1, 1-q^k)^2*(1-q^n)) ) ) \\ Joerg Arndt, Dec 09 2012
(Sage)
def A001522(n):
if n < 4: return 1
return (number_of_partitions(n) - [p.crank() for p in Partitions(n)].count(0))/2
[A001522(n) for n in range(30)] # Peter Luschny, Sep 15 2014
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
a(0) changed from 0 to 1 by Joerg Arndt, Mar 30 2014
Edited definition. - N. J. A. Sloane, Mar 31 2021
STATUS
approved