 They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Dudley E. Littlewood (1961).The Hall–Littlewood polynomial "P" is defined by
where λ is a partition of at most "n" with elements λ"i", and "m"("i") elements equal to "i", and "S""n" is the symmetric group of order "n"!.As an example,

We have that formula_3, formula_4 and 
formula_5 where the latter is the Schur "P" polynomials.