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A005987 Number of symmetric plane partitions of n.
(Formerly M0562)
12
1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 41, 53, 71, 93, 125, 160, 211, 270, 354, 450, 581, 735, 948, 1191, 1517, 1902, 2414, 3008, 3791, 4709, 5909, 7311, 9119, 11246, 13981, 17178, 21249, 26039, 32105, 39213, 48159, 58669, 71831, 87269 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

From M. F. Hasler, Sep 26 2018: (Start)

A plane partition of n is a matrix of nonnegative integers that sum up to n, and such that A[i,j] >= A[i+1,j], A[i,j] >= A[i,j+1] for all i,j. We can consider A of infinite size but there are at most n nonzero rows and columns and we ignore empty rows or columns. It is symmetric iff A = transpose(A), i.e., A[i,j] = A[j,i] for all i,j.

For any n, we have A000219(n) = a(n) + 2*A306098(n) where A306098(n) is the number of equivalence classes, modulo transposition, of non-symmetric plane partitions. (For any of these, its transpose is a different plane partition of n.) (End)

REFERENCES

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 134.

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. 2, 1999; see Corollary 7.20.5

LINKS

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

A. Björner and R. P. Stanley, with  A combinatorial miscellany, L'Enseignement Math., Monograph No. 42, 2010.

R. P. Stanley, Theory and application of plane partitions II, Studies in Appl. Math., 50 (1971), 259-279. DOI:10.1002/sapm1971503259. [Scan on author's personal web page].

FORMULA

G.f.: Product_{i=1..oo} 1/(1-x^(2i-1))/(1-x^(2i))^floor(i/2). (Stanley 1971, Prop.14.3; Björner & Stanley 2010, p. 33).

a(n) ~ exp(3 * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (2^(10/3) * Zeta(3)^(1/3)) - Pi^4 / (384*Zeta(3)) + 1/24) * Zeta(3)^(13/72) / (2^(77/72) * sqrt(3*Pi*A) * n^(49/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 05 2018

EXAMPLE

From M. F. Hasler, Sep 26 2018: (Start)

The only plane partition of n = 0 is the empty partition []; we consider it to be symmetric (as a 0 X 0 matrix), so a(0) = 1.

The only plane partition of n = 1 is the partition [1] which is symmetric, so a(1) = 1.

For n = 2 we have the partitions [2], [1 1] and [1; 1] (where ; denotes the end of a row). Only the first one is symmetric, so a(2) = 1.

For n = 3 we have the partitions [3], [2 1], [2; 1], [1 1; 1 0], [1 1 1], [1; 1; 1]. The first and the fourth are symmetric, so a(3) = 2. (End)

MATHEMATICA

terms = 46; s = Product[1/(1 - x^(2i-1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[terms/2]}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 10 2017 *)

PROG

(PARI) a(n)=polcoeff(prod(k=1, n, (1-x^k)^-if(k%2, 1, k\4), 1+x*O(x^n)), n) \\ Michael Somos, May 19 2000

(PARI) show(n)=select(t->(t=matconcat(t~))~==t, PlanePartitions(n)) \\ Using PlanePartitions() given in A091298, this selects and returns the list of symmetric plane partitions of n. - M. F. Hasler, Sep 26 2018

CROSSREFS

Cf. A000784, A000785, A000786, A000219, A048142.

Sequence in context: A321729 A180652 A046682 * A241828 A125895 A241344

Adjacent sequences:  A005984 A005985 A005986 * A005988 A005989 A005990

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Wouter Meeussen, Dec 11 1999

Edited by M. F. Hasler, Sep 26 2018

STATUS

approved

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)