A partial recurrence relation for reduced class coefficients of the symmetric group

An expression for the product of a single‐cycle class [(1) N ‐ P ( p )] N and an arbitrary class [(1) l 1 (2) l … ( N ) lN ] N of the symmetric group has recently been conjectured. This expression involves a sum over a relatively small number of reduced class sums, depending on p indices. A further conjecture is formulated and demonstrated, according to which reduced class coefficients ( RCCS ) involving cycles whose length is expressed by means of a single index can be related to corresponding coefficients in the product of [(1) N ‐ P +1 ( p ‐ 1)] N with an arbitrary class sum. Consequently, the problem of evaluating the general class sum product reduces to that of obtaining a relatively small set of fundamental RCCS containing no single‐index cycles. The conjectures mentioned can be used to evaluate the product [(1) N ‐ p ( p )] N · [(1) N ‐ q ( q )] N in terms of fundamental RCCS that can all be obtained from the product [( r )] r · [( r )] r , where r = min( p , q ). For the latter product, we use a result due to Boccara.