

A005157


Number of totally symmetric plane partitions that fit in an n X n X n box.
(Formerly M1499)


7



1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, 1885102080, 62062015500, 2652584509440, 147198472495020, 10606175914819584, 992340657705109416, 120567366227960791040, 19023173201224270401428, 3897937005297330777227264
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Also, number of 2dimensional shifted complexes on n+1 nodes. [Klivans]
Also the number of totally symmetric partitions which fit in an (n1)dimensional box with side length 4 (for n>0).  Graham H. Hawkes, Jan 11 2014
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 ddimensional box with side length n is equal to the number of totally symmetric partitions which fit inside an ndimensional box of side length d.  Graham H. Hawkes, Jan 11 2014
For two other contexts where this sequence arises, see the Knuth (2019) link (noncrossing paths among the 2(2^n1) paths defined in that note; independent sets of paths among the first 2^n1 of those).  N. J. A. Sloane, Feb 09 2019, based on email from Don Knuth.


REFERENCES

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198 (corrected).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..130 (first 51 terms from T. D. Noe)
R. K. Guy, Letter to N. J. A. Sloane, Dec 5 1988.
Graham H. Hawkes, Totally symmetric partitions in boxes
C. Klivans, Obstructions to shiftedness, preprint
C. Klivans, Obstructions to shiftedness, Discrete Comput. Geom., 33 (2005), 535545.
Don Knuth, A conjecture about noncrossing paths, Feb 06 2019
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]


FORMULA

a(n) = Product_{i=1..n, Product_{j=i..n, Product_{k=j..n, (i+j+k1)/(i+j+k2)}}}.  Paul Barry, May 13 2008
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 GlaisherKinkelin constant.  Vaclav Kotesovec, Mar 01 2015
a(n) = sqrt(A323848(n+1,n)) for n >= 1. [proof by Nikolai Beluhov; see Knuth (2019) link]  Alois P. Heinz, Feb 10 2019
Apparently, a(n) = Sum_{k=0..n} A184173(n,k).  Alois P. Heinz, Feb 11 2019


EXAMPLE

a(2) = 5 because we have: void, 1, 21/1, 22/21, and 22/22.


MAPLE

A005157 := proc(n) local i, j; mul(mul((i+j+n1)/(i+2*j2), j=i..n), i=1..n); end;


MATHEMATICA

Table[Product[(i+j+k1)/(i+j+k2), {i, n}, {j, i, n}, {k, j, n}], {n, 0, 20}] (* Harvey P. Dale, Jul 17 2011 *)


PROG

(PARI) A005157(n)=prod(i=1, n, prod(j=i, n, (i+j+n1)/(i+2*j2))) \\ M. F. Hasler, Sep 26 2018


CROSSREFS

Cf. A049505, A184173, A323848.
Row sums of A214564.
Sequence in context: A091139 A084785 A124551 * A019502 A019503 A019504
Adjacent sequences: A005154 A005155 A005156 * A005158 A005159 A005160


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



