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平面劃分

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平面劃分是整數 n_(i,j) 的二維陣列，從左到右和從上到下都是非遞增的，並且總和為給定的數字 n。換句話說，

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)

這個定義的隱含要求是陣列頂部和左側對齊，並且不包含空洞。

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

例如，上面說明了 22 的一個平面劃分。

生成函式，用於平面劃分的數量 PL(n)，為 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)。

寫作 a(n)=PL(n)， 遞推方程，用於 a(n)，由下式給出

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

其中 sigma_k(n) 是除數函式。它也有生成函式

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

MacMahon (1960) 還表明，Young tableaux 楊氏表 適合 a×b 矩形內部且整數不超過 c 的平面劃分的數量 PL(a,b,c) （換句話說，所有 n_(i,j)<=c）由下式給出

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)。展開乘積得到

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-函式。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)

其中前幾項是 2, 20, 980, 232848, 267227532, 1478619421136, ... (OEIS A008793)。令人驚訝的是，PL(a,b,c) 也給出了邊長為 a, b, c, a, b, c 的六邊形由菱形鋪砌的數量 (David and Tomei 1989, Fulmek and Krattenthaler 2000)。

平面劃分的概念也可以推廣到立方劃分。

另請參閱

迴圈對稱平面劃分, 降序平面劃分, 六邊形鋪砌, 劃分, Macdonald 平面劃分猜想, 立體劃分, 完全對稱自互補平面劃分, Young Tableau

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參考文獻

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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平面劃分

引用為

Weisstein, Eric W. “平面劃分。” 來自 網路資源。 https://mathworld.tw/PlanePartition.html

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