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%I M0562 #45 Sep 27 2018 12:32:08
%S 1,1,1,2,3,4,6,8,12,16,22,29,41,53,71,93,125,160,211,270,354,450,581,
%T 735,948,1191,1517,1902,2414,3008,3791,4709,5909,7311,9119,11246,
%U 13981,17178,21249,26039,32105,39213,48159,58669,71831,87269
%N Number of symmetric plane partitions of n.
%C From _M. F. Hasler_, Sep 26 2018: (Start)
%C 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.
%C 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)
%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 134.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Corollary 7.20.5
%H T. D. Noe, <a href="/A005987/b005987.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Björner and R. P. Stanley, with <a href="http://www-math.mit.edu/~rstan/papers/comb.pdf">A combinatorial miscellany</a>, L'Enseignement Math., Monograph No. 42, 2010.
%H R. P. Stanley, <a href="http://doi.org/10.1002/sapm1971503259">Theory and application of plane partitions II</a>, Studies in Appl. Math., 50 (1971), 259-279. DOI:10.1002/sapm1971503259. <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/12-2.pdf">[Scan on author's personal web page]</a>.
%F 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).
%F 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
%e From _M. F. Hasler_, Sep 26 2018: (Start)
%e 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.
%e The only plane partition of n = 1 is the partition [1] which is symmetric, so a(1) = 1.
%e 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.
%e 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)
%t 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 *)
%o (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
%o (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
%Y Cf. A000784, A000785, A000786, A000219, A048142.
%K nonn,nice,easy
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Wouter Meeussen_, Dec 11 1999
%E Edited by _M. F. Hasler_, Sep 26 2018
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