Self-conjugate $$(s,s+d,\dots ,s+pd)$$-core partitions and free Motzkin paths

Combinatorics on bounded free Motzkin paths and its applications

“…by applying Case 1-ii) and Case 0, we get 11,3), by applying Case 2-ii), we obtain (7), where P 1 is the path given in (a). See Figure 5 for further details.…”

“…Although the number of self-conjugate (t, t + 1, • • • , t + p)-cores with the fixed number of corners is unknown in general, it is enumerated in [2,3] when p = 1, 2, and 3. The number of self-conjugate t-core partitions with m corners can be counted by using these path interpretations.…”

Combinatorics on Bounded Free Motzkin Paths and its Applications

“…Although the number of self-conjugate (t, t + 1, • • • , t + p)-cores with the fixed number of corners is unknown in general, it is enumerated in [2,3] when p = 1, 2, and 3. The number of self-conjugate t-core partitions with m corners can be counted by using these path interpretations.…”

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

“…For instance, Figure 1 gives the Young diagram and hook lengths of the partition (6, 3, 2, 1) and Figure 2 gives the Young diagram and hook lengths of its conjugation. The partition (6, 3, 2, 1) is a (4,6,11)-core partition since its hook length set doesn't contain multiples of 4, 6 or 11. One can note the first column hook lengths are 9, 5, 3, 1 and they uniquely determine the partition.…”

“…Simultaneous core partitions are connected with rational combinatorics (see [16]). 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]).…”