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A001523 Number of stacks, or planar partitions of n; also weakly unimodal compositions of n.
(Formerly M1102 N0420)
110
1, 1, 2, 4, 8, 15, 27, 47, 79, 130, 209, 330, 512, 784, 1183, 1765, 2604, 3804, 5504, 7898, 11240, 15880, 22277, 31048, 43003, 59220, 81098, 110484, 149769, 202070, 271404, 362974, 483439, 641368, 847681, 1116325, 1464999, 1916184, 2498258 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) counts stacks of integer-length boards of total length n where no board overhangs the board underneath.

Number of graphical partitions on 2n nodes that contain a 1. E.g. a(3)=4 and so there are 4 graphical partitions of 6 that contain a 1, namely (111111), (21111), (2211) and (3111). Only (222) fails. - Jon Perry, Jul 25 2003

It would seem from Stanley that he regards a(0)=0 for this sequence and A001522. - Michael Somos, Feb 22 2015

In the article by Auluck is a typo in the formula (24), 2*Pi is missing in an exponent on the left side of the equation for Q(n). The correct term is exp(2*Pi*sqrt(n/3)), not just exp(sqrt(n/3)). - Vaclav Kotesovec, Jun 22 2015

REFERENCES

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

T. D. Noe and Alois P. Heinz, 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, g(x).

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)

H. Bottomley, Illustration of initial terms

Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.

Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46

R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms

A. D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011.

Eric Weisstein's World of Mathematics, Unimodal Sequence

E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

FORMULA

a(n) = Sum_{k=1..n} f(k, n-k), where f(n, k) (= A054250) = 1 if k = 0; Sum_{j=1..min(n, k)} (n-j+1) f(j, k-j)) if k > 0. - David W. Wilson, May 05 2000

a(n) = Sum_{k} A059623(n, k) for n > 0. - Henry Bottomley, Feb 01 2001

A006330(n) + a(n) = A000712(n). - Michael Somos, Jul 22 2003

G.f.: 1 + (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2. - Michael Somos, Jul 22 2003

G.f.: 1 + Sum_{n>=1} (x^n / ( ( Product_{k=1..n-1} (1 - x^k)^2 ) * (1-x^n) ) ). - Joerg Arndt, Oct 01 2012

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015

a(n) + A115981(n) = 2^(n - 1). - Gus Wiseman, Mar 04 2020

EXAMPLE

For a(4)=8 we have the following stacks:

x

x x. .x

x x. .x x.. .x. ..x xx

x xx xx xxx xxx xxx xx xxxx

G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 27*x^6 + 47*x^7 + 79*x^8 + ...

From Gus Wiseman, Mar 04 2020: (Start)

The a(1) = 1 through a(5) = 15 unimodal compositions:

  (1)  (2)   (3)    (4)     (5)

       (11)  (12)   (13)    (14)

             (21)   (22)    (23)

             (111)  (31)    (32)

                    (112)   (41)

                    (121)   (113)

                    (211)   (122)

                    (1111)  (131)

                            (221)

                            (311)

                            (1112)

                            (1121)

                            (1211)

                            (2111)

                            (11111)

(End)

MAPLE

b:= proc(n, i) option remember;

      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+

      add(b(n-i*j, i+1)*(j+1), j=0..n/i))

    end:

a:= n-> `if`(n=0, 1, b(n, 1)):

seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

MATHEMATICA

max = 40; s = 1 + Sum[(-1)^(k + 1)*q^(k*(k + 1)/2), {k, 1, max}] / QPochhammer[q]^2 + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Jan 25 2012, updated Nov 29 2015 *)

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i]==0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 0, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 24 2015, after Alois P. Heinz *)

unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], unimodQ[#]&]], {n, 0, 10}] (* Gus Wiseman, Mar 04 2020 *)

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))^2 , n))}; /* Michael Somos, Jul 22 2003 */

(Python)

def b(n, i):

    if i>n: return 0

    if n%i==0: x=1

    else: x=0

return x + sum([b(n - i*j, i + 1)*(j + 1) for j in range(n/i + 1)])

def a(n): return 1 if n==0 else b(n, 1) # Indranil Ghosh, Jun 09 2017, after Maple code by Alois P. Heinz

CROSSREFS

Cf. A054250, A059618, A059623, A001522, A001524.

Cf. A000569. Bisections give A100505, A100506.

Row sums of A247255.

Row sums of A072704.

The strict case is A072706.

The complement is counted by A115981.

The case covering an initial interval is A227038.

The version whose negation is unimodal as well appears to be A329398.

Unimodal sequences covering an initial interval are A007052.

Non-unimodal permutations are A059204.

Non-unimodal sequences covering an initial interval are A328509.

Partitions with unimodal run-lengths are A332280.

Numbers whose prime signature is not unimodal are A332282.

Partitions whose 0-appended first differences are unimodal are A332283.

The number of unimodal permutations of the prime indices of n is A332288.

Compositions whose negation is unimodal are A332578.

Compositions whose run-lengths are unimodal are A332726.

Cf. A107429, A156253, A332285, A332294, A332577, A332642, A332669.

Sequence in context: A331554 A222038 A328087 * A222039 A222148 A222040

Adjacent sequences:  A001520 A001521 A001522 * A001524 A001525 A001526

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson, May 05 2000

Definition corrected by  Wolfdieter Lang, Dec 05 2018

STATUS

approved

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Last modified March 4 11:25 EST 2021. Contains 341791 sequences. (Running on oeis4.)