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A001523 Number of stacks, or planar partitions of n; also weakly unimodal partitions of n.
(Formerly M1102 N0420)
+0
15
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; listen)
OFFSET

0,3

COMMENT

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

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

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 (perry(AT)globalnet.co.uk), Jul 25 2003

REFERENCES

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs. Proc. Cambridge Philos. Soc. 47, (1951), 679-686.

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

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

H. Bottomley, Illustration of initial terms

FORMULA

a(n) = Sum(1 <= k <= n, f(k, n-k)), where f(n, k) (=A054250) = 1 if k = 0; Sum(1 <= j <= min(n, k); (n-j+1) f(j, k-j)) if k > 0.

a(n)=sum_k[A059623(n, k)] for n>0 - Henry Bottomley (se16(AT)btinternet.com), Feb 01 2001

G.f.: (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2.

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

PROGRAM

(PARI) a(n)=if(n<1, 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))

CROSSREFS

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

Cf. A000569. Bisections give A100505, A100506.

Adjacent sequences: A001520 A001521 A001522 this_sequence A001524 A001525 A001526

Sequence in context: A003241 A125513 A054174 this_sequence A000126 A098057 A074029

KEYWORD

nonn,nice,easy

AUTHOR

njas

EXTENSIONS

Formula and more terms from David W. Wilson (davidwwilson(AT)comcast.net) May 05 2000.

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Last modified May 16 23:01 EDT 2008. Contains 139884 sequences.


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