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A008793 The problem of the calissons: number of ways to tile hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box. 31
1, 2, 20, 980, 232848, 267227532, 1478619421136, 39405996318420160, 5055160684040254910720, 3120344782196754906063540800, 9265037718181937012241727284450000, 132307448895406086706107959899799334375000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The 3-dimensional analog of A000984. - William Entriken, Aug 06 2013

REFERENCES

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545, also p. 575 line -1 with a=b=c=n.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198. The first printing of Eq. (6.8) is wrong (see A049505 and A005157), but if one changes the limits in the formula (before it is corrected) to {1 <= i <= r, 1 <= j <= r}, one obtains the present sequence. - N. J. A. Sloane, Jun 30 2013

Gordon G. Cash and Jerry Ray Dias, Computation, Properties and Resonance Topology of Benzenoid Monoradicals and Polyradicals and the Eigenvectors Belonging to Their Zero Eigenvalues, J. Math. Chem., 30 (2001), 429-444. [See K, p. 442.]

Sebastien Desreux, Martin Matamala, Ivan Rapaport, Eric Remila, Domino tilings and related models: space of configurations of domains with holes, arXiv:math/0302344, 27 Feb 2003

Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, Martin van Hecke1, Topological defects produce exotic mechanics in complex metamaterials, arXiv preprint 1903.07919, 2019 [See Section "Compatible metamaterials with fully antiferromagnetic interactions" - N. J. A. Sloane, Mar 23 2019]

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see p. 261).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..54 (terms 0..30 from T. D. Noe)

T. Amdeberhan and V. H. Moll, Arithmetic properties of plane partitions, El. J. Comb. 18 (2) (2011) # P1.

Guy David and Carlos Tomei, The Problem of the Calissons, The American Mathematical Monthly, Vol. 96, No. 5 (May, 1989), pp. 429-431 (3 pages).

P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.

I. Fischer, Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center, arXiv:math/9906102 [math.CO], 1999.

P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.

M. Fulmek and C. Krattenthaler, The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II, arXiv:math/9909038 [math.CO], 1999.

I. Gutman, S. J. Cyvin, and V. Ivanov-Petrovic, Topological properties of circumcoronenes, Z. Naturforsch., 53a, 1998, 699-703 (see p. 700) - Emeric Deutsch, May 14 2018

H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.

Sam Hopkins and Tri Lai, Plane partitions of shifted double staircase shape, arXiv:2007.05381 [math.CO], 2020. See Table 1 p. 9.

C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO], 2005.

P. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.

Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, Martin van Hecke, Topological defects produce exotic mechanics in complex metamaterials, arXiv:1903.07919 [cond-mat.soft], 2019.

James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

James Propp, Updated article

James Propp, Tiling Problems, Old and New, Rutgers University Math Colloquium, March 30, 2022

N. C. Saldanha and C. Tomei, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.

P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.

Eric Weisstein's World of Mathematics, Plane Partition.

FORMULA

a(n) = Product_{i = 0..n-1} (i^(-i)*(n+i)^(2*i-n)*(2*n+i)^(n-i)).

a(n) = Product_{i = 1..n} Product_{j = 0..n-1} (3*n-i-j)/(2*n-i-j).

a(n) = Product_{i = 1..n} Gamma[i]*Gamma[i+2*n]/Gamma[i+n]^2.

a(n) = Product_[i = 0..n-1} i!*(i+2*n)!/(i+n)!^2 .

a(n) = Product_{i = 1..n} Product_{j = n..2*n-1} i+j / Product_{j = 0..n-1} i+j. - Paul Barry, Jun 13 2006

For n >= 1, a(n) = det(binomial(2*n,n+i-j)) for 1<=i,j<=n [Krattenhaller, Theorem 4, with a = b = c = n].

Let H(n) = Product_{k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Chapter II, Section 429, p. 182, with x -> 1]. Setting a = b = c = n gives the entries for this sequence. - Peter Bala, Dec 22 2011

a(n) ~ exp(1/12) * 3^(9*n^2/2 - 1/12) / (A * n^(1/12) * 2^(6*n^2 - 1/4)), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 27 2015

a(n) = Product_{i = 1..n} Product_{j = 1..n} (n+i+j-1)/(i+j-1). - Michel Marcus, Jul 13 2020

Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1))^p (mod p^(4*r)) hold for all primes p and positive integers n and r. - Peter Bala, Apr 07 2022

MAPLE

A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;

MATHEMATICA

Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]

PROG

(PARI) a(n) = prod(i=1, n, prod(j=1, n, (n+i+j-1)/(i+j-1))); \\ Michel Marcus, Jul 13 2020

CROSSREFS

Cf. A000984, A066931, A352656, A352657. Main diagonal of array A103905.

Sequence in context: A135757 A301945 A158843 * A015192 A012790 A273194

Adjacent sequences:  A008790 A008791 A008792 * A008794 A008795 A008796

KEYWORD

nonn,easy,nice

AUTHOR

Jonas Wallgren

EXTENSIONS

More terms from Eric W. Weisstein

STATUS

approved

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Last modified May 17 12:38 EDT 2022. Contains 353746 sequences. (Running on oeis4.)