Partitions with modular multiplicities

Abstract We study the partition function p ( d ; n ), defined as the number of partitions of n in which each part occurs a number of times congruent to 0 or 1 modulo d , for integers $$d\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We derive a compact generating function and introduce a dual signed function a ( d ; n ) whose reciprocal generating series leads to convolution identities expressing a form of combinatorial orthogonality between the two families. These identities yield recurrence relations and explicit formulas involving divisor sums. Using the modular structure of the generating functions, we establish several Ramanujan-type congruences for p (5; n ) and p (7; n ), as well as congruences for a (7; n ). We also investigate the parity of p ( d ; n ) for prime values of $$d\ge 5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> , obtaining infinite families of parity congruences governed by quadratic character conditions.