TOTALITY OF PARTS IN ALL PARTITIONS OF AN INTEGER

International Journal of Advanced Trends in Engineering and Technology (IJATET) International Peer Reviewed - Refereed Research Journal, Website: www.dvpublication.com Impact Factor: 5.965, ISSN (Online): 2456 - 4664, Volume 10, Issue 2, July - December, 2025 112 TOTALITY OF PARTS IN ALL PARTITIONS OF AN INTEGER R. Sivaraman*, G. Sánchez-Meléndez** & J. López-Bonilla** * Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai, Tamil Nadu, India ** ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 5, 1er. Piso, Col. Lindavista CP 07738, CDMX, México Cite This Article: R. Sivaraman, G. Sánchez-Meléndez & J. López-Bonilla, “Totality of Parts in All Partitions of an Integer”, International Journal of Advanced Trends in Engineering and Technology, Volume 10, Issue 2, July - December, Page Number 112-113, 2025. Copy Right: © DV Publication, 2025 (All Rights Reserved). This is an Open Access Article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited. DOI: Abstract: We employ the Merca and Uchimura theorems to obtain an expression for the totality of parts in all partitions of an integer. Key Words: Merca‟s Theorem, Integer Partitions, Divisor Function, Uchimura‟s Theorem. 1. Introduction: Merca [1-3] deduced the following expression for an arbitrary arithmetic function f n : f k n K=1 Sn,k = g k n k=1 p n−k , g m = f d , d/m (1) Where Sn,k is the number of k‟s in all partitions of n, and p(m) is the partition function [4]. For the case f = e, that is, f k = 1, then g m is the divisor function d(m) [4] and thus (1) gives the totality of parts in all partitions of the integer n: Sn,k n k=1 = d(k) n k=1 p n−k . (2) On the other hand, if Qj(n) is the number of partitions of n into (possibly repeated) parts from among {1, 2, …, j}, its generating function is given by [5]: Qj(n) ∞ n=0 qn=1 (q ; q)j , (3) Such that lim⁡ j→∞Qj n = p n , that is: p(n) ∞ n=0 qn=1 (q ; q)∞ . (4) In Sec. 2 we use a theorem of Uchimura [5, 6] to deduce a connection between (2) and the Qj n . 2. All Partitions of an Integer and Their Totality of Parts: The Uchimura‟s theorem gives the following relation [5, 6]: 1 (q ; q)∞ d(n) ∞ n=0 qn= j (q ; q)j ∞ j=0 qj , (5) Where we can apply (3) and (4) to obtain: ( ∞ n=0 d j n j=0 p(n −j)) qn= j ∞ j=0 Qj m ∞ m=0 qj+m = j ∞ j=0 Qj(n −j) ∞ n=j qn, = j n j=0 ∞ n=0 Qj n−j qn , Therefore: Sn,k n K=1 = d k n k=1 p n−k = k n k=1 Qk n−k , (6) Is an interesting connection between the Qm(n) and the total number of parts in all partitions of an integer. Remark: Qk(n) also is the number of partitions of n into at most k parts [7]. References: 1. M. Merca, A note on a classical connection between partitions and divisors, Ann. Acad. Rom. Sci. Ser. Math. Appl. 15, No. 1-2 (2023) 163-174. 2. B. E. Sagan, Rooted partitions and number-theoretic functions, arXiv: 2319.19098v1 [math.NT] 29 Oct 2023. 3. M. Alegri, J. Prajapati, J. López-Bonilla, On the Merca‟s connection between the partition function and Euler‟s totient, Revista Sergipana de Matemática (REVISEM, Brazil) 9, No. 3 (2024) 136-143. 4. R. Sivaramakrishnan, Classical theory of arithmetic functions, Marcel Dekker, New York, USA (1989). 5. W. P. Johnson, An introduction to q-analysis, Am. Math. Soc., Providence, Rhode Island, USA (2020). 6. Hei-Chi Chan, An invitation to q-series. From Jacobi‟s triple product identity to Ramanujan‟s „most beautiful identity‟, World Scientific, Singapore (2011). International Journal of Advanced Trends in Engineering and Technology (IJATET) International Peer Reviewed - Refereed Research Journal, Website: www.dvpublication.com Impact Factor: 5.965, ISSN (Online): 2456 - 4664, Volume 10, Issue 2, July - December, 2025 113 7. R. P. Stanley, Enumerative combinatorics. I, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1999).