A005157 Number of totally symmetric plane partitions that fit in an n X n X n box.

1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, 1885102080, 62062015500, 2652584509440, 147198472495020, 10606175914819584, 992340657705109416, 120567366227960791040, 19023173201224270401428, 3897937005297330777227264

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Also, number of 2-dimensional shifted complexes on n+1 nodes. [Klivans]

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

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

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.

Examples

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

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

• Alois P. Heinz, Table of n, a(n) for n = 0..130 (first 51 terms from T. D. Noe)
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• R. K. Guy, Letter to N. J. A. Sloane, Dec 5 1988.
• Graham H. Hawkes, Totally symmetric partitions in boxes
• Seth Ireland, A bijection between strongly stable and totally symmetric partitions, arXiv:2302.02505 [math.CO], 2023.
• C. Klivans, Obstructions to shiftedness, preprint.
• C. Klivans, Obstructions to shiftedness, Discrete Comput. Geom., 33 (2005), 535-545.
• Don Knuth, A conjecture about noncrossing paths, Feb 06 2019.
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Crossrefs

Cf. A049505, A184173, A323848.

Row sums of A214564.

Programs

Maple

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

Mathematica

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 *)

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

Formula

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

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

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

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