Congruences for the number of kk-tuple partitions with distinct even parts

Let k≥1k≥1 be an integer and pedk(n)pedk(n) be the number of kk-tuple partitions of nn wherein even parts are distinct (and odd parts are unrestricted). We prove a class of congruences for pedk(n)pedk(n) mod 2 by Hecke nilpotency.