

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
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OFFSET

0,4


COMMENTS

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 nonsymmetric 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



FORMULA

G.f.: Product_{i=1..oo} 1/(1x^(2i1))/(1x^(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 GlaisherKinkelin constant A074962.  Vaclav Kotesovec, May 05 2018


EXAMPLE

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^(2i1))/(1  x^(2i))^Floor[i/2], {i, 1, Ceiling[terms/2]}] + O[x]^terms; CoefficientList[s, x] (* JeanFrançois Alcover, Jul 10 2017 *)


PROG

(PARI) a(n)=polcoeff(prod(k=1, n, (1x^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



KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS



STATUS

approved



