Arithmetic properties of restricted colour partition functions Bk(n) and Ck(n)

S. Biswas, N. Saikia

For any positive integer n and even integer k>0, let Bk(n) is the number of partitions of n such that no part is congruent to zero modulo 2k, parts congruent to even integers other than k modulo 2k have one colour and parts congruent to odd integers and k modulo 2k have two colours. Also, for any odd integer k>0, let Ck(n) is the number of partitions of an integer n>0 such that no parts congruent to zero modulo 2k, parts congruent to k modulo 2k have three colours, parts congruent to even integers modulo 2k have one colour and parts congruent to odd integers other than k modulo 2k have two colours. In this paper, we establish some congruences modulo powers of 2 for some particular cases of k associated with the functions Bk(n) and Ck(n). Some recurrence relations connecting the functions with certain partition functions are also established. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 19(1) (2026), pp. 45-63