Parabolic
R-polynomials were introduced by Deodhar as parabolic analogues of ordinary
R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic
R -polynomials for the symmetric group. Let
0fa" title="Click to view the MathML source">Sn be the symmetric group on
{1,2,…,n}, and let
![View the MathML source View the MathML source](/sd/grey_pxl.gif)
be the generating set of
0fa" title="Click to view the MathML source">Sn, where for
1≤i≤n−1,
si is the adjacent transposition. For a subset
J⊆S, let
f697f546828c6f758" title="Click to view the MathML source">(Sn)J be the parabolic subgroup generated by
J , and let
f69187fd7813e6cf431e93" title="Click to view the MathML source">(Sn)J be the set of minimal coset representatives for
Sn/(Sn)J. For
u≤v∈(Sn)J in the Bruhat order and
x∈{q,−1}, let
20" width="53" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300780-si12.gif"> denote the parabolic
R-polynomial indexed by
u and
v . Brenti found a formula for
20" width="53" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300780-si12.gif"> when
J=S∖{si}, and obtained an expression for
20" width="53" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300780-si12.gif"> when
J=S∖{si−1,si}. In this paper, we provide a formula for
20" width="53" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300780-si12.gif">, where
J=S∖{si−2,si−1,si} and
i appears after
i−1 in
v. It should be noted that the condition that
i appears after
i−1 in
v is equivalent to that
v is a permutation in
0f0e37e0" title="Click to view the MathML source">(Sn)S∖{si−2,si}. We also pose a conjecture for
20" width="53" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300780-si12.gif">, where
J=S∖{sk,sk+1,…,si} with
1≤k≤i≤n−1 and
v is a permutation in
20" class="mathmlsrc">20.gif&_user=111111111&_pii=S0022404916300780&_rdoc=1&_issn=00224049&md5=c99bd72a4a3c29196cb8bf4f62f83f5c" title="Click to view the MathML source">(Sn)S∖{sk,si}.