Quarter-factorials (just three examples to show all apparent behaviours ...) 

 

 

 

p = 5(the order values appear to go: gamma, `+`(`*`(2, `*`(gamma, `*`(p)))), `*`(`^`(p, 2)), `+`(`*`(2, `*`(gamma, `*`(`^`(p, 3))))), `*`(`^`(p, 4)), `+`(`*`(2, `*`(gamma, `*`(`^`(p, 5))))), `...` 

 

 

 

p = 13(the order values appear to go: gamma, `*`(gamma, `*`(p)), `*`(`^`(p, 2)), `*`(gamma, `*`(`^`(p, 3))), `*`(`^`(p, 4)), `*`(gamma, `*`(`^`(p, 5))), () .. () 

 

 

 

p = 37(the order values appear to go: gamma, `+`(`*`(`/`(1, 2), `*`(gamma, `*`(p)))), `*`(`^`(p, 2)), `+`(`*`(`/`(1, 2), `*`(gamma, `*`(`^`(p, 3))))), `*`(`^`(p, 4)), `+`(`*`(`/`(1, 2), `*`(gamma, `*`(`^`(p, 5))))), `...`