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A001522 Number of n-stacks with strictly receding walls, or planar partitions of n.
(Formerly M0644 N0238)
14
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 (list; graph; refs; listen; history; text; internal format)
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

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)

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

a(n) = (A000041(n) - A064410(n)) / 2 for n>=2.

G.f.: 1 + sum(k>=1, -(-1)^k * x^(k*(k+1)/2) ) / prod(k>=1, 1-x^k ).

G.f.: 1 + sum(n>=1, q^(n^2) / ( prod(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 *)

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

Cf. A000041, A059776, A001523, A001524.

Sequence in context: A280277 A102108 A105780 * A054405 A155167 A237269

Adjacent sequences:  A001519 A001520 A001521 * A001523 A001524 A001525

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(0) changed from 0 to 1 by Joerg Arndt, Mar 30 2014

STATUS

approved

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Last modified July 27 20:56 EDT 2017. Contains 289866 sequences.