GENERALIZATION OF UNIVERSAL PARTITION AND BIPARTITION THEOREMS. Hacène Belbachir 1

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1 #A59 INTEGERS 3 (23) GENERALIZATION OF UNIVERSAL PARTITION AND BIPARTITION THEOREMS Hacène Belbachir USTHB, Faculty of Mathematics, RECITS Laboratory, Algiers, Algeria hbelbachir@usthb.dz, hacenebelbachir@gmail.com Miloud Mihoubi 2 USTHB, Faculty of Mathematics, RECITS Laboratory, Algiers, Algeria. mmihoubi@usthb.dz, miloudmihoubi@gmail.com Received: 5/24/2, Revised: 5/4/3, Accepted: 8//3, Published: 9/26/3 Abstract Let A = (a i,j ), i =, 2,..., j =,, 2,..., be an infinite matrix with elements a i,j = or ; p (n, k; A) the number of partitions of n into k parts whose number y i of parts which are equal to i belongs to the set i = {j : a i,j = }, i =, 2,.... The universal theorem on partitions states that n p (n, k; A) u k t n a i,j u j t ij A. n=k= In this paper, we present a generalization of this result. We show that this generalization remains true when a i,j are indeterminate. We also take into account the bi-partite and multi-partite situations. i= j=. Introduction Let A = (a i,j ), i =, 2,..., j =,, 2,... be an infinite matrix with elements a i,j = or ; p (n, k; A) the number of partitions of n into k parts whose number y i of parts which are equal to i belongs to the set i = {j : a i,j = }, i =, 2,.... The universal theorem on partitions states that n p (n, k; A) u k t n a i,j u j t ij A ; () n= k= The research is partially supported by the LITIS Laboratory of Rouen University and the PNR project 8/u6/ This research is supported by the PNR project 8/u6/372. i= j=

2 INTEGERS: 3 (23) 2 see for instance [2] and [3]. In Section 2, we will provide an extension of the above identity and show that it remains true when a i,j are indeterminate. In Section 3, we will present an equivalent version in terms of complete Bell polynomials when a i, =, i. Similarly, a partition of an ordered pair (m, n) 6= (, ), of nonnegative integers, is a non-ordered collection of nonnegative integers (x i, y i ) 6= (, ), i =, 2,..., whose sum equals (m, n). Given a partition of (m, n), let k i,j be the number of parts which are equal to (i, j), i =,, 2,..., m, j =,, 2,..., n, (i, j) 6= (, ), such that m n n m i k i,j = m, j k i,j = n. (2) i= j= For a partition of (m, n) into k parts, we add m n k i,j = k. (3) i= j= Let p (m, n) be the number of partitions of the bi-partite number (m, n) with p (, ) = and p (m, n, k) be the number of partitions of (m, n) into k parts with p (,, ) =. The universal bipartition theorem states that F (t, u, w) = m,n j= i= p (m, n, k) w k t m u n = k= j=i= (i,j)6=(,) wt i u j ; (4) see [2, p. 43, pb. 24]. A generalization of identity (4) is dealt with in Section 4. Section 5 is devoted to the concept of multipartition. 2. Generalized Universal Partition Theorem Theorem. Let = (x i,j ), i =, 2,..., j =,, 2,..., be an infinite matrix of indeterminates; (n, k) the set of all nonnegative integer solutions of k + k k n = k and k + 2k nk n = n; and (n) = S n k= (n, k) be the set of all nonnegative integer solutions of k + 2k nk n = n. For every solution k, k 2,..., k n, we set p (n, k; ) := x,k x 2,k2 x n,kn. Then G (t, u; ) := n= k= (n,k) n p (n, k; ) u k t n x i,j u j t ij A. i= j=

3 INTEGERS: 3 (23) 3 Proof. We have G (t, u; ) = n=k= x,k x 2,k2 x n,kn u k+ +kn t k+2k2+ +nkn A. (n,k) Since these sums apply for all k =,,..., n and n =,,..., it follows that n G (t, u; ) x,k (ut) x k 2,k2 ut 2 k2 x n,kn (ut n ) kn A n=k= (n,k) n= (n) = x,k (ut) k = k i= j= x,k (ut) k x 2,k2 ut 2 k2 x i,j ut i j which is the required expression. A, x 2,k2 ut 2 k2 k 2= x n,kn (ut n ) kn A For x i,j = a i,j with i =, 2,..., j =,, 2,..., in Theorem, we obtain the universal theorem on partitions. For x i,j = ai,j ji with i =, 2,..., j =,, 2,..., in j Theorem, we obtain: Corollary 2. Let A = (a i,j ), i =, 2,..., j =,, 2,..., be an infinite matrix with elements a i,j = or and c (n, k; A) the number of permutations of a finite set W n, of n elements, that are decomposed into k cycles such that the number of cycles of length i belongs to the set i = {j : a i,j = }. Then n c (n, k; A) u k tn n u j t i j a i,j A. j i n=k= For x i,j = z i j j i ai,j, z i 2 C, i =, 2,..., j =,, 2,..., in Theorem, we obtain a remarkable identity according to the partial Bell polynomials: Corollary 3. Let A = (a i,j ), i =, 2,..., j =,, 2,..., be an infinite matrix with elements a i,j = or and B n,k;a (z, z 2,..., z n ) := (n,k) i= n k k n j= z k zn kn a,k a n,kn. n

4 INTEGERS: 3 (23) 4 Then we obtain n B n,k;a (z, z 2,..., z n ) u k tn n n=k= i= j= a i,j j t uz i j i A. i Remark 4. For a i,j =, i =, 2,... and j =,,..., Corollary 3 gives n B n,k (z, z 2,..., z n ) u k tn n = exp u t i z i, i n=k= which is the definition of the partial Bell polynomials B n,k (z,..., z n ). See [, 3, 4]. i= 3. Connection With the Complete Bell Polynomials Recall that the complete Bell polynomials A n (x, x 2,... ) are defined by A n (x, x 2,... ) tn n = exp t m x m. m See [, 3, 4]. n=k In this section, we provide another formulation of Theorem according to the complete Bell polynomials. We determine the generating functions of the sequences (p (n, k; )) n and (p (n, k; )) k, where = (x i,j ), i =, 2,..., j =,, 2,..., is an infinite matrix with indeterminates x i,j such that x i, = for every i. m= Theorem 5. Let q, u be indeterminate. Then, for n, we have p (j, n; ) q j = n A n ( (q; ), 2 (q; ),..., n (q; )), (5) j=n n p (n, j; ) u j = n A n ( (u; ), 2 (u; ),..., n (u; )), (6) j= where with n u n (q; ) := n (i) q i=b ni k and n (u; ) := n b k k k, k n b n (i) = n ( ) k (k )B n,k (x i,, 2x i,2,..., jx i,j,... ). k=

5 INTEGERS: 3 (23) 5 Proof. From Theorem, we get n G (q, u; ) := p (n, k; ) u k q n = + (jx i,j ) uqi j AA. j n=k= i= j= Using the following known expansion (see [2, Theorem.7]) q k q n ln + g k = c n k n, k= with c n = P n k= ( )k (k )B n,k (g, g 2,... ), we obtain G (q, u; ) = exp b k (i) uk u k k qki = exp b k (i) q ki k i=k= k= i= = exp k (q; ) uk k k= = + A k ( (q; ),..., k (q; )) uk k. On the other hand, we have G (q, u; ) = n=k= n= k= n p (n, k; ) u k q n = k= p (n, k; ) q n u k. The first identity follows from identification, where as the second identity follows from the expansion n G (q, u; ) = p (n, k; ) u k q n n= k= n=k = + (jx i,j ) uqi j AA j i= j= = exp b k (i) u k qik k i=k= = n=q n n u k b k A k k k n = exp n (u; ) qn n n= = + A n ( (u; ), 2 (u; ),..., n (u; )) qn n. n= 2

6 INTEGERS: 3 (23) 6 Corollary 6. Let and q be two indeterminates. We then have A n q n,..., (n ) q n n = n q n i= q i and A ) u,..., (n ku n/k ( u) n AA = n (p n (u) up n (u)), k n P where p n (u) := n p (n, j) u j. j= Proof. We put in identity (5) x, =, x,j = q j ( ) for j and x i,j = q j (n ) n for i 2, j, and use identity B n,k (, 2, 3,... ) = (k ) k (Lah numbers). We obtain b n () = (n ) ( n ) q n, b n (i) = (n )q n, i 2, n (q; ) = b n (i) q ni = (n ) q n n, i= p (n, k; ) = q k + q k ( ) (n,k), k = = q k (n k,k) (n,k), k + ( ) q k (n,k ) = q k [p (n k, k) + ( ) p (n, k )], where p (n, k) is the number of partitions of n into k parts, which satisfy p (n, k) = p (n k, k) + p (n, k ). Thus, we obtain p (j, n; ) q j = q p (j, n) q j j=n = j=n n q n i= p (j, n ) q j A j=n q i, which gives the first identity. For the second identity, we take x, =, x,j = for j and x i,j = for

7 INTEGERS: 3 (23) 7 i 2, j, in relation (6) to get b n () = (n ) ( n ), b n (i) = (n ), i 2, n (u; ) = n n u k b k k k = (n ) ( u)n + (n ) ku n/k, k n k n p (n, k; ) = p (n, k) p (n, k ), thus n p (n, j; ) u j = p n (u) up n (u), j= which provides the second identity. 4. Generalized Universal Bipartition Theorem In this section, we provide a generalization of identity (4) and deduce some known identities. Let us start with the following example: how do we partition (2, 3) into di erent parts? Let p (2, 3, k) be the number of partitions of the bi-partite number (2, 3) into k parts, k =,..., 5 and p (2, 3) be the total number of partitions of (2, 3). We have (2, 3) p (2, 3, ) = (, ) + (2, 2) = (, 2) + (2, ) = (, 3) + (2, ) = (, ) + (, 3) = (, ) + (, 2) p (2, 3, 2) = 5 (, ) + (, ) + (2, ) = (, ) + (, 2) + (2, ) = (, ) + (, ) + (, ) = (, ) + (, ) + (, 2) = (, 3) + (, ) + (, ) p (2, 3, 3) = 5 (, ) + (, ) + (, ) + (2, ) = (, ) + (, ) + (, ) + (, ) = (, ) + (, ) + (, ) + (, 2) p (2, 3, 4) = 3 (, ) + (, ) + (, ) + (, ) + (, ) p (2, 3, 5) = p (2, 3) = 5 P k= p (2, 3, k) = = 5. Theorem 7. Let = (x i,j,s ), i, j, s =,, 2,..., be a sequence of indeterminates with x,,s =, (m, n, k) the set of all nonnegative integers k i,j satisfying (2) and (3) and (m, n) := S n+m k= (m, n, k) the set of all nonnegative integers satisfying (2). For every partition of the bi-partite number (m, n) into k parts, we set p (m, n, k; ) := m n x i,j,ki,j. (m,n,k) i=j=

8 INTEGERS: 3 (23) 8 Then we have F (t, u, ; ) := m,n p (m, n, k; ) k t m u n = k= i+j x i,j,s t i u j s. s= Proof. We have F (t, u, ; ) = m,n = m,n k= = p (m, n, k; ) k t m u n m,n (m,n) = (m,n,k) mp P i n np P k i,j j m mp np k i,j k i,j m i= j= u j= i= i=j= t m x i,j,ki,j t i u j ki,j A i=j= x i,j,ki,j t i u j ki,j A. k i,j= n i=j= x i,j,ki,j A Corollary 8. Let A = (a i,j,s ), i, j, s =,, 2,..., with a i,j,s = or for (i, j) 6= (, ) and let p (m, n, k; A) be the number of partitions of (m, n) into k parts whose number y i,j of parts which are equal to (i, j) belongs to the set i,j = {s : a i,j,s = }, i, j =,, 2,..., (i, j) 6= (, ). Then F (t, u, ; A) := p (m, n, k; A) k t m u n = a i,j,s t i u j s. m,n k= i+j For x i,j,s = for all i, j, s =,, 2,..., (i, j) 6= (, ), Theorem 7 becomes: Corollary 9. Let p (m, n) be the number of partitions of the bi-partite number (m, n) with p (, ) = and p (m, n, k) the number of partitions of (m, n) into k parts, with p (,, ) =. Then p (m, n, k) k t m u n = t i u j. m,n k= i+j Remark. Let (y i,j ), i, j =,,..., be a sequence of indeterminates and let s x i,j,s = yi,j s ij, i, j =,, 2,..., we have p (m, n, k; ) = m n k i,j (m,n,k) i=j= ki,j yi,j ij s= = A m,n,k, mn

9 INTEGERS: 3 (23) 9 where and A m,n,k := A m,n,k (y,, y,,..., y m,n ), F (t, u, ; ) = i+j k i,j m,n A m,n,k k k= t m m u n n. From Theorem 7, we obtain F (t, u, ; ) t y i u j ki,j i,j A t i u j y i,j AA. k i,j i j i j i+j From the two expressions of F (t, u;, ), we retrieve the exponential partial bipartitional polynomials: A m,n,k (y,, y,,..., y m,n ) k t m u n m n t i u j y i,j AA ; i j m,n k= see [2, pp: ]. i+j 5. Universal Multipartition Theorem More generally, a multipartition of order r of n = (n,..., n r ), di erent from = (,..., ), of nonnegative integers, is a non-ordered collection of nonnegative integers x () i,..., x (r) i, i =, 2,..., whose sum equals n. In a partition of an r-partite number n, let k i := k i,...,i r be the number of ordered r numbers that are equal to i = (i,..., i r ) 2 {,, 2,..., n } {,, 2,..., n r }, (i,..., i r ) 6=, such that n n r i j k i,...,i r = n j, j =,..., r. (7) i = i r= For the partition of n into k parts, we add n i = n r k i,...,i r = k. (8) i r= Let p (n) be the number of partitions of the r-partite n with p () = and p (n, k) the number of partitions of the r-partite number n into k parts, with p (, ) =. Theorem. Let = (x i,s ), i = (i,..., i r ) 2 N r, i 6=, s =,, 2,..., be a sequence of indeterminates with r + indices, with x,s =, (n, k) the set of all nonnegative integers k i,...,i r satisfying (7) and (8) and (n) := S n + +n r k= (n, k)

10 INTEGERS: 3 (23) the set of all nonnegative integers solutions of (7). For every partition of n into k parts, we set p (n; k, ) := n x i,ki. Then F (t, ; ) = n (n,k) i= n p (n, k; ) k t n = k= i x i,s t i s. where t n := t n... tnr r, n : = n + + n r, n, n,..., n r. Proof. We have the following s= F (t, ; ) = n p (n, k; ) k t n n k= = n P i x i,ki j applen k i r j t n k= (n,k) i= j= = n x i,ki t i k i, n (n) i= P i j applen i j jk i j and exploiting x, =, the last expression i,ki ki x t i k ia i,ki ki x t i k ia. i i A For x i,s = a i,s 2 {, } for all i 2 N r, i 6=, s =,, 2,..., we obtain: Corollary 2. Let A = (a i,s ), i 2 N r, i 6=, s =,, 2,..., with a i,s = or and p (n, k; A) be the number of partitions of n into k parts whose number y i of parts which are equal to i belongs to the set i = {s : a i,s = }, i 2 N r, i 6=. Then F (t, u, ; A) := n p (n, k; A) k t n a i,s t i s A. n s k= i For x i,s = for all i 2 N r, i 6=, s =,, 2,..., we obtain:

11 INTEGERS: 3 (23) Corollary 3. Let p (n) be the number of partitions of the r-partite number n with p () = and p (n, k) the number of partitions of the r-bipartite number n into k parts, with p (, ) =. Then n p (n, k) k t n = t i. Consequently n k= p (n) t n = n i i t i. Remark 4. If we take t = = t r = t, we obtain t i ( i+r r ) n = i n k= = n n + +n r=n n + +n r=n p n () p (n, k) t n, k t n and more generally, for nonnegative integers a,..., a r and t = t a,..., t r = t ar, we obtain t i f(i,r) = p n () t n, where i p n () = n a n + +a rn r k= a n + +a rn r=n p (n, k) k, and where f (n, r) is the number of solutions of the integer equation a n + a 2 n a r n r = n. Acknowledgments The authors wish to express their gratitude to the referee for his/her valuable advice and comments which helped to greatly in improving the quality of the paper. References [] E. T. Bell, Exponential polynomials. Ann. Math. 35 (934), [2] Charalambos A. Charalambides, Enumerative Combinatorics. Chapman & Hall/CRC, A CRC Press Company, Boca Raton, London, New ork, Washington, D.C. (2). [3] L. Comtet, Advanced Combinatorics. D. Reidel Publishing Company, Dordrecht-Holland / Boston-U.S.A, (974). [4] M. Mihoubi, Bell polynomials and binomial type sequences. Discrete Math., 38 (28),