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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A001522 Number of n-stacks with strictly receding walls, or the number of Type A partitions of n in the sense of Auluck (1951).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 47, 62, 82, 107, 139, 179, 230, 293, 372, 470, 591, 740, 924, 1148, 1422, 1756, 2161, 2651, 3244, 3957, 4815, 5844, 7075, 8545, 10299, 12383, 14859, 17794, 21267, 25368, 30207, 35902, 42600, 50462, 59678, 70465, 83079, 97800, 114967, 134956, 158205, 185209, 216546, 252859
Offset: 0

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Author

N. J. A. Sloane

Keywords

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

Examples

			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 + ...

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

Crossrefs

Cf. A000041, A059776, A001523, A001524.
A version for permutations is A002467, complement A000166.
The case of zero crank is A064410, ranked by A342192.
The case of nonnegative crank is A064428, ranked by A352873.
A strict version is A352829, complement A352828.
Conjectured to be column k = 1 of A352833.
These partitions (positive crank) are ranked by A352874.
A000700 counts self-conjugate partitions, ranked by A088902.
A064391 counts partitions by crank.
A115720 and A115994 count partitions by their Durfee square.
A257989 gives the crank of the partition with Heinz number n.
Counting compositions: A003242, A114921, A238351, A342527, A342528, A342532.
Fixed points of reversed partitions: A238352, A238394, A238395, A352822, A352830, A352872.
Cf. A008292, A114088, A118199, A325187, A352827, A352832.

Programs

  • 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 *)
  • 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

Formula

a(n) = (A000041(n) - A064410(n)) / 2 for n>=2.
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
a(n) = A000041(n) - A064428(n). - Gus Wiseman, Mar 30 2021
a(n) = A064428(n) - A064410(n). - Gus Wiseman, May 23 2022

Extensions

a(0) changed from 0 to 1 by Joerg Arndt, Mar 30 2014
Edited definition. - N. J. A. Sloane, Mar 31 2021

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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