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Volume 11 (2024) | Issue 5
Volume 11 (2024) | Issue 5
Volume 11 (2024) | Issue 5
Volume 11 (2024) | Issue 5
Volume 11 (2024) | Issue 5
Abstract. Lin introduced and studied two new partition functions ???????????? (????) and ???????????????? (????), which count the number of designated summands in two partition functions ????????(????) and ????????????(????), respectively. In this paper, we prove the generating functions for ???????????????? (????????????), ????????????????(???????????? + ????????), ???????????????? (???????????? + ????????), ????????????????(???????????? + ????????) and ????????????????(???????????? + ????????). Also, we find some new congruences and infinite families of congruences modulo 16 and modulo 72 for ???????????????? (????).