Lower bounds of 4, 3, and 8 additional relations for T_n, I_n, PT_n beyond the symmetric group, with matching explicit presentations for n large enough.
Concerning the abstract groups of order k! and (1/2)k! holohedrically isomorphic with the symmetric and the alternating substitution-groups on k letters
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Short presentations for transformation monoids
Lower bounds of 4, 3, and 8 additional relations for T_n, I_n, PT_n beyond the symmetric group, with matching explicit presentations for n large enough.