Formulas for the number of k -colored partitions and the number of plane partitions of n in terms of the Bell polynomials
aa r X i v : . [ m a t h . G M ] S e p FORMULAS FOR THE NUMBER OF k -COLORED PARTITIONS ANDTHE NUMBER OF PLANE PARTITIONS OF n IN TERMS OF THEBELL POLYNOMIALS
SUMIT KUMAR JHA
Abstract.
We derive closed formulas for the number of k -coloured partitions and thenumber of plane partitions of n in terms of the Bell polynomials. Introduction
Definition 1 ( k -colored partitions of n [1]) . A partition of a positive integer n is a finiteweakly decreasing sequence of positive integers λ ≥ λ ≥ · · · ≥ λ r > P ri =1 λ i = n . The λ i ’s are called the parts of the partition. Let p ( n ) denote the number of partitionsof n .A partition is said to be k -coloured if each part can occur as k colours. Let p k ( n ) denotethe number of k coloured partitions of n . For example, there are five 2-coloured partitionsof 2: 1 + 1 , ′ + 1 , ′ + 1 ′ , , ′ so p (2) = 5.The generating function for p k ( n ) is given by ∞ X n =0 p k ( n ) q n = ∞ Y j =1 − q j ) k ( | q | < . The right hand side of the above equation is also the generating function for the multi-partitions which appear in many papers on the representations theory of Lie algebras, andsome with applications in physics [2].The unrestricted partition function p ( n ) = p ( n ) can be computed using HardyRamanu-janRademacher formula with soft complexity O ( n / o (1) ) and very little overhead [3].In section 2, we obtain a formula for the number of k -colored partitions of n in terms ofthe partial Bell polynomials which allows computing them efficiently. For example, usingour formula we can calculate p (200) = 23945275792616100703623332622769220026826156718318470749445535353589which are the number of thirty colored partitions of 200. Mathematics Subject Classification.
Key words and phrases. k -coloured partition; plane partitions. Definition 2 (Plane partitions of n ) . A plane partition of n is a two-dimensional array ofintegers n i,j that are non-increasing in both indices, that is, n i,j ≥ n i,j +1 n i,j ≥ n i +1 ,j and n = X i,j n i,j . Let PL ( n ) denote the number of plane partitions of n . For example, there are six planepartitions of 3: 1 1 1 1 11 111 2 1 21 3so PL (3) = 6.Macmohan [4] obtained the generating function of PL ( n ) as ∞ X n =0 PL ( n ) q n = ∞ X j =1 − q j ) j ( | q | < . In section 3, we obtain an expression for PL ( n ) in terms of the complete Bell polynomialsand sum of squares of divisors of n . This also gives a determinant expression for PL ( n ).For example, we can calculate PL (700) = 1542248695905922088013690041381656661664744761954709483748320717869 . Definition 3 (Partial Bell polynomials) . For n and k non-negative integers, the partial( n, k )th partial Bell polynomials in the variables x , x , . . . , x n − k +1 denoted by B n,k ≡ B n,k ( x , x , . . . , x n − k +1 ) [5, p. 206] can be defined by B n,k ( x , x , . . . , x n − k +1 ) = X ≤ i ≤ n,ℓ i ∈ N P ni =1 iℓ i = n P ni =1 ℓ i = k n ! Q n − k +1 i =1 ℓ i ! n − k +1 Y i =1 (cid:16) x i i ! (cid:17) ℓ i . The partial Bell polynomials have the following generating function:1 k ! ∞ X j =1 x j t j j ! ! k = ∞ X n = k B n,k ( x , . . . , x n − k +1 ) t n n ! , k = 0 , , , . . .. The partial Bell polynomials can also be computed efficiently by a recurrence relation [6]: B n,k = n − k +1 X i =1 (cid:18) n − i − (cid:19) x i B n − i,k − , where B , = 1 OLORED AND PLANE PARTITIONS 3 B n, = 0 for n ≥ B ,k = 0 for k ≥ . Cvijovi´c [7] gives the following formula for calculating these polynomials B n,k +1 = 1( k + 1)! n − X α = k α − X α = k − · · · α k − − X α k =1 | {z } k k z }| {(cid:18) nα (cid:19)(cid:18) α α (cid:19) · · · (cid:18) α k − α k (cid:19) · x n − α x α − α · · · x α k − − α k x α k ( n ≥ k + 1 , k = 1 , , . . . ) (1)The n th complete Bell polynomials are defined as B n ( x , x , · · · , x n ) = n X k =0 B n,k ( x , x , . . . , x n − k +1 ) . The complete Bell polynomials can be recurrently defined as B n +1 ( x , . . . , x n +1 ) = n X i =0 (cid:18) ni (cid:19) B n − i ( x , . . . , x n − i ) x i +1 with the initial value B = 1. We would frequently use Fa`a di Bruno’s formula [5, p. 134]which is d n dq n f ( g ( q )) = n X l =1 f ( l ) ( g ( q )) · B n,l (cid:0) g ′ ( q ) , g ′′ ( q ) , . . . , g ( n − l +1) ( q ) (cid:1) . (2)2. Formula for k -coloured partitions of n Theorem 1.
For all positive integers n, kp k ( n ) = 1 n ! n X l =0 ( − l ( k ) ( l ) B n,l ( λ , λ , · · · , λ n − l +1 ) where λ i = ( − m i ! if i = m (3 m +1)2 for some positive integer m ( − m i ! if i = m (3 m − for some positive integer m , where ( k ) ( l ) = k ( k + 1) · · · ( k + l − represents the rising factorial.Proof. We recall the Euler’s pentagonal number theorem [9, Equation 7.8] g ( q ) := ∞ Y j =1 (1 − q j ) = ∞ X n = ∞ ( − n q n n . SUMIT KUMAR JHA
Let f ( q ) = q − k . Then the Fa`a di Bruno’s formula (2) gives d n dq n g ( q ) − k = n X l =1 ( − l ( k ) ( l ) ( g ( q )) l + k B n,l (cid:0) g ′ ( q ) , g ′′ ( q ) , . . . , g ( n − l +1) ( q ) (cid:1) . Letting q → (cid:3) Another expression for p k ( n ) Theorem 2.
For all positive integers n, kp k ( n ) = B n ( k σ (1) , k σ (2) , · · · , k ( n − σ ( n )) n ! , where σ ( n ) = P d | n d .Proof. Let h ( q ) := ∞ Y j =1 − q j ) k . Then we have log h ( q ) = − ∞ X j =1 k log(1 − q j )= ∞ X j =1 ∞ X l =1 k q lj l = ∞ X n =1 k q n X d | n d − . = ∞ X n =1 q n kn X d | n d. Let f ( q ) = e q and g ( q ) = log h ( q ) in Fa`a di Bruno’s formula (2) gives d n dq n h ( q ) = n X l =1 h ( q ) B n,l (cid:0) g ′ ( q ) , g ′′ ( q ) , . . . , g ( n − l +1) ( q ) (cid:1) . Letting q → (cid:3) OLORED AND PLANE PARTITIONS 5
The determinant expression for the complete Bell polynomials [8, Theorem 2.1] gives us p k ( n ) = 1 n ! det k σ (1) k σ (2) k σ (3) k σ (4) · · · · · · k σ ( n ) − k σ (1) k σ (2) k σ (3) · · · · · · k σ ( n − − k σ (1) k σ (2) · · · · · · k σ ( n − − k σ (1) · · · · · · k σ ( n − − · · · · · · k σ ( n − · · · − ( n − k σ (1) . Thus p (2) = 12 det (cid:20) − (cid:21) = 5 . Formula for Plane partitions of n Theorem 3.
For all positive integers n PL ( n ) = B n ( σ (1) , σ (2) , · · · , ( n − σ ( n )) n ! , where σ ( d ) = P d | n d .Proof. Let h ( q ) := ∞ Y j =1 − q j ) j . Then we have log h ( q ) = − ∞ X j =1 j log(1 − q j )= ∞ X j =1 ∞ X l =1 j q lj l = ∞ X n =1 n q n X d | n d − . = ∞ X n =1 q n n X d | n d . SUMIT KUMAR JHA
Let f ( q ) = e q and g ( q ) = log h ( q ) in Fa`a di Bruno’s formula (2) gives d n dq n h ( q ) = n X l =1 h ( q ) B n,l (cid:0) g ′ ( q ) , g ′′ ( q ) , . . . , g ( n − l +1) ( q ) (cid:1) . Letting q → (cid:3) We immediately have the following determinant formula for PL ( n ) [8, Theorem 2.1] PL ( n ) = 1 n ! det σ (1) σ (2) σ (3) σ (4) · · · · · · σ ( n ) − σ (1) σ (2) σ (3) · · · · · · σ ( n − − σ (1) σ (2) · · · · · · σ ( n − − σ (1) · · · · · · σ ( n − − · · · · · · σ ( n − · · · − ( n − σ (1) . For example, using the above we have PL (3) = 16 det − − = 6 . References [1] S. Chern, S. Fu & D. Tang, Some inequalities for k -colored partition functions, Ramanujan J 46,713–725 (2018).[2] Andrews G.E. (2008) A Survey of Multipartitions Congruences and Identities. In: Surveys in Num-ber Theory. Developments in Mathematics (Diophantine Approximation: Festschrift for WolfgangSchmidt), vol 17. Springer, New York, NY.[3] Johansson, F. (2012). Efficient implementation of the HardyRamanujanRademacher formula. LMSJournal of Computation and Mathematics, 15, 341-359.[4] Knuth, D. (1970). A Note on Solid Partitions. Mathematics of Computation, 24(112), 955-961[5] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions , D. Reidel PublishingCo., Dordrecht, 1974.[6] H. Taghavian, A fast algorithm for computing Bell polynomials based on index break-downs usingprime factorization, preprint, 2020. Available at https://arxiv.org/abs/2004.09283 .[7] D. Cvijovi´c, New identities for the partial Bell polynomials, Applied mathematics letters, (2011),1544–1547.[8] A. Xu & Z. Cen, On a q -analogue of Fa`a di Brunos determinant formula, Discrete Mathematics, (2011), 387–392.[9] N. J. Fine, Basic Hypergeometric Series and Applications , American Mathematical Soc., 1988.
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