The size of Bruhat intervals between nested involutions in SnSn

We build a chainid=ϑ0<ϑ1<⋯<ϑ⌊n2⌋−1<ϑ⌊n2⌋ of nested involutions in the Bruhat ordering of SnSn, with ϑ⌊n2⌋ the maximal element for the Bruhat order, and we study the cardinality of the Bruhat intervals [ϑj,ϑk][ϑj,ϑk] for all 0⩽j