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%I M1499 #100 Oct 27 2023 19:30:30
%S 1,2,5,16,66,352,2431,21760,252586,3803648,74327145,1885102080,
%T 62062015500,2652584509440,147198472495020,10606175914819584,
%U 992340657705109416,120567366227960791040,19023173201224270401428,3897937005297330777227264
%N Number of totally symmetric plane partitions that fit in an n X n X n box.
%C Also, number of 2-dimensional shifted complexes on n+1 nodes. [Klivans]
%C Also the number of totally symmetric partitions which fit in an (n-1)-dimensional box with side length 4 (for n>0). - _Graham H. Hawkes_, Jan 11 2014
%C Suppose we index this sequence slightly differently. Let the elements of a partition be represented by points rather than boxes, as in a Ferrers diagram. In this case, a 1 X 1 X 1 (closed) box would fit 8 points -— one at each vertex of the box, and we use the convention that a 0 X 0 X 0 (closed) box contains exactly one point. Using this indexing, the sequence begins (offset is still 0) 2,5,16,... rather than 1,2,5,... If we use the same method of indexing for all other dimensions, then we have the following remarkable result: The number of totally symmetric partitions which fit inside a d-dimensional box with side length n is equal to the number of totally symmetric partitions which fit inside an n-dimensional box of side length d. - _Graham H. Hawkes_, Jan 11 2014
%C For two other contexts where this sequence arises, see the Knuth (2019) link (noncrossing paths among the 2(2^n-1) paths defined in that note; independent sets of paths among the first 2^n-1 of those). - _N. J. A. Sloane_, Feb 09 2019, based on email from _Don Knuth_.
%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198 (corrected).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A005157/b005157.txt">Table of n, a(n) for n = 0..130</a> (first 51 terms from T. D. Noe)
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H R. K. Guy, <a href="/A005157/a005157_3.pdf">Letter to N. J. A. Sloane</a>, Dec 5 1988.
%H Graham H. Hawkes, <a href="/A005157/a005157_2.pdf">Totally symmetric partitions in boxes</a>
%H Seth Ireland, <a href="https://arxiv.org/abs/2302.02505">A bijection between strongly stable and totally symmetric partitions</a>, arXiv:2302.02505 [math.CO], 2023.
%H C. Klivans, <a href="http://www.dam.brown.edu/people/cklivans/obs_cjk.pdf">Obstructions to shiftedness</a>, preprint.
%H C. Klivans, <a href="http://dx.doi.org/10.1007/s00454-004-1103-9">Obstructions to shiftedness</a>, Discrete Comput. Geom., 33 (2005), 535-545.
%H Don Knuth, <a href="https://cs.stanford.edu/~knuth/papers/noncr-conj.pdf">A conjecture about noncrossing paths</a>, Feb 06 2019.
%H R. P. Stanley, <a href="http://dx.doi.org/10.1007/BFb0072521">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
%H R. P. Stanley, <a href="/A005130/a005130.pdf">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]
%F a(n) = Product_{i=1..n} Product_{j=i..n} Product_{k=j..n} (i+j+k-1)/(i+j+k-2). - _Paul Barry_, May 13 2008
%F a(n) ~ exp(1/72) * GAMMA(1/3)^(2/3) * n^(7/72) * 3^(3*n*(n+1)/4 + 11/72) / (A^(1/6) * Pi^(1/3) * 2^(n*(2*n+1)/2 + 13/24)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Mar 01 2015
%F a(n) = sqrt(A323848(n+1,n)) for n >= 1. [proof by Nikolai Beluhov; see Knuth (2019) link] - _Alois P. Heinz_, Feb 10 2019
%F Apparently, a(n) = Sum_{k=0..n} A184173(n,k). - _Alois P. Heinz_, Feb 11 2019
%F Conjectures: if p == 1 (mod 6) is prime then a(p) == 2^((p+5)/6) (mod p^2); if p == 5 (mod 6) is prime then a(p) == 2^((p+1)/6) (mod p^2) (checked up to p = 1009). - _Peter Bala_, Feb 17 2023
%e a(2) = 5 because we have: void, 1, 21/1, 22/21, and 22/22.
%p A005157 := proc(n) local i,j; mul(mul((i+j+n-1)/(i+2*j-2),j=i..n),i=1..n); end;
%t Table[Product[(i+j+k-1)/(i+j+k-2),{i,n},{j,i,n},{k,j,n}],{n,0,20}] (* _Harvey P. Dale_, Jul 17 2011 *)
%o (PARI) A005157(n)=prod(i=1,n,prod(j=i,n,(i+j+n-1)/(i+2*j-2))) \\ _M. F. Hasler_, Sep 26 2018
%Y Cf. A049505, A184173, A323848.
%Y Row sums of A214564.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_
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