“…Also, simultaneous core partitions are connected with Motzkin paths and Dyck paths (see [9,11,39,40]). Some statistics of simultaneous core partitions, such as numbers of partitions, numbers of corners, largest sizes and average sizes, have attracted much attention in the past twenty years (see [2,3,4,7,8,10,13,14,15,18,20,21,23,24,25,29,31,33,35,37,42,43,44]). For example, Anderson [3] showed that the number of (s 1 , s 2 )-core partitions is equal to 1 s 1 +s 2 s 1 +s 2 s 1…”