Proof of Conjecture 19 of Ballantine, Beck, Merca, and Sagan on Elementary Symmetric Partitions

Ballantine, Beck, Merca, and Sagan conjectured four identities, collectively Conjecture 19, relating the image of the map pre_k on integer partitions to four OEIS sequences. We prove parts (i) and (iii) unconditionally, prove part (iv) unconditionally using the injectivity of pre_2 on partitions of n (Conjecture 1 of the same paper, proved by Li in arXiv:2508.00971), and show that this injectivity is in fact equivalent to part (iv). For part (ii) we prove the partition-theoretic half unconditionally and reduce the remaining content to a 2006 conjecture of Dean Hickerson on the OEIS concerning Huffman coding. We also correct a sign error in the published statement of part (iii): the correct identity is chi(ImP_3(n)) = A213213(n) - 1, not 1 + A213213(n) as stated.