I've heard this before, but not in the Army. I can't find it posted at Mudcat:

OLD KING COLE (CADENCE COUNT)

Old King Cole was a merry old soul and a merry old soul was he
He called for his pipe and called for his bowl
And he called for his privates, three
Beer, Beer, Beer said the privates
Merry men are we
But none so fair that they can compare to the airborne infantry (alt: fighting infantry)

Old King Cole was a merry old soul and a merry old soul was he
He called for his pipe and he called for his bowl
and he called for his corporals, three
I need a three-day pass said the corporals
Beer, Beer, Beer said the privates
Merry men are we
But none so fair that they can compare to the airborne infantry

Additional Verses:
Sergeants three–“File from the left, column right said the Sgts”
Lieuues three–“I’ll lead the way said the lieuies”
Captains three–“Charge that hill said the captains”
Majors three–“Who’s gonna shine my boots said the majors”
Colonels three–“Where’s my star said the colonels”
Generals three–“War, War, War”

Source: https://www.armystudyguide.com/content/Prep_For_Basic_Training/Prep_for_basic_general_information/common-marching-cadence.shtml

Any other versions?