The immanant 
of an 
matrix 
, indexed by a partition 
of 
, is the weighted sum
 |
where 
is the irreducible group
character of the symmetric group 
indexed by 
(Littlewood and Richardson 1934a). The partitions 
and 
give the permanent and determinant, respectively.
For 
and 
, the dimension
of the corresponding irreducible representation
is 
, where 
is the identity
permutation. Skandera (2026, p. 16) proved that for 
every 
totally nonnegative
matrix satisfies
 |
The hypothesis requires all minors to be nonnegative, rather than only the entries. The inequalities above concern the specified partitions.
