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平面划分

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平面划分是整数 n_(i,j) 的二维数组，从左到右和从上到下都是非递增的，并且总和为给定的数字 n。换句话说，

n_(i,j)>=n_(i,j+1)
(1)
n_(i,j)>=n_(i+1,j)
(2)

和

n=sum_(i,j)n_(i,j).
(3)

这个定义的隐含要求是数组顶部和左侧对齐，并且不包含空洞。

PlanePartition
5 4 2 1 1; 3 2 ; 2 2
(4)

例如，上面说明了 22 的一个平面划分。

生成函数，用于平面划分的数量 PL(n)，为 n，是

sum_(n=0)^inftyPL(n)x^n=1/(product_(k=1)^(infty)(1-x^k)^k)=1+x+3x^2+6x^3+13x^4+24x^5+...
(5)

(OEIS A000219, MacMahon 1912b, Speciner 1972, Bender and Knuth 1972, Bressoud and Propp 1999)。

写作 a(n)=PL(n)， 递推方程，用于 a(n)，由下式给出

a(n)=1/nsum_(k=1)^na(n-k)sigma_2(k),
(6)

其中 sigma_k(n) 是除数函数。它也有生成函数

G(x)=exp[sum_(n=1)^inftysigma_2(n)(x^n)/n].
(7)

MacMahon (1960) 还表明，Young tableaux 杨氏表 适合 a×b 矩形内部且整数不超过 c 的平面划分的数量 PL(a,b,c) （换句话说，所有 n_(i,j)<=c）由下式给出

PL(a,b,c)=product_(i=1)^aproduct_(j=1)^bproduct_(k=1)^c(i+j+k-1)/(i+j+k-2)
(8)

(Bressoud and Propp 1999, Fulmek and Krattenthaler 2000)。展开乘积得到

PL(a,b,c)=product_(i=1)^(a)(Gamma(i)Gamma(b+c+i))/(Gamma(b+i)Gamma(c+i))
(9)
=(G(a+1)G(b+1)G(c+1)G(a+b+c+1))/(G(a+b+1)G(a+c+1)G(b+c+1)),
(10)

其中 G(n) 是 Barnes G-函数。取 n=a=b=c 得到

PL(n,n,n)=product_(i=1)^(n)(Gamma(i)Gamma(i+2n))/([Gamma(i+n)]^2)
(11)
=([G(n+1)]^3G(3n+1))/([G(2n+1)]^3),
(12)

其中前几项是 2, 20, 980, 232848, 267227532, 1478619421136, ... (OEIS A008793)。令人惊讶的是，PL(a,b,c) 也给出了边长为 a, b, c, a, b, c 的六边形由菱形铺砌的数量 (David and Tomei 1989, Fulmek and Krattenthaler 2000)。

平面划分的概念也可以推广到立方划分。

另请参阅

循环对称平面划分, 降序平面划分, 六边形铺砌, 划分, Macdonald 平面划分猜想, 立体划分, 完全对称自互补平面划分, Young Tableau

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参考文献

Bender, E. A. and Knuth, D. E. "Enumeration of Plane Partitions." J. Combin. Theory Ser. A. 13, 40-54, 1972.Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999.Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637-646.Cohn, H.; Larsen, M.; and Propp, J. "The Shape of a Typical Boxed Plane Partition." New York J. Math. 4, 137-166, 1998.David, G. and Tomei, C. "The Problem of the Calissons." Amer. Math. Monthly 96, 429-431, 1989.Fulmek, M. and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry Axes, II." Europ. J. Combin. 21, 601-640, 2000.Knuth, D. E. "A Note on Solid Partitions." Math. Comput. 24, 955-961, 1970.MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. V: Partitions in Two-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211, 75-110, 1912a.MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. VI: Partitions in Two-Dimensional Space, to which is Added an Adumbration of the Theory of Partitions in Three-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211, 345-373, 1912b.MacMahon, P. A. §429 and 494 in Combinatory Analysis, Vol. 2. New York: Chelsea, 1960.Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr. "Proof of the Macdonald Conjecture." Invent. Math. 66, 73-87, 1982.Sloane, N. J. A. Sequences A000219/M2566 and A008793 in "The On-Line Encyclopedia of Integer Sequences."Speciner, M. Item 18 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 10, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item18.Stanley, R. P. "Symmetry of Plane Partitions." J. Combin. Th. Ser. A 3, 103-113, 1986.Stanley, R. P. "A Baker's Dozen of Conjectures Concerning Plane Partitions." In Combinatoire Énumérative: Proceedings of the "Colloque De Combinatoire Enumerative," Held at Université Du Quebec a Montreal, May 28-June 1, 1985 (Ed. G. Labelle and P. Leroux). New York: Springer-Verlag, 285-293, 1986.

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平面划分

引用为

Weisstein, Eric W. “平面划分。” 来自 网络资源。 https://mathworld.net.cn/PlanePartition.html

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