The descent set of an SRCT [tau] of size n, denoted by Des([tau]), is
Given a composition [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.k]), the canonical tableau of shape [alpha], denoted by [[tau].sub.[alpha]], is the unique SRCT of shape [alpha] and comp([[tau].sub.[alpha]]) = ([[alpha].sub.1], ..., [[alpha].sub.k]).
yields the following SRCT [tau] with Des([tau]) = {1,3,4,5,8}.
Example 4.3 Let [tau] be the SRCT of shape (3,4,2,3) shown below.
The proof involves an intricate case analysis of the action of [[pi].sub.i] on an SRCT [tau] depending on whether or not i [member of] Des([tau]).
Definition 5.1 Let [tau] be an SRCT of shape [alpha] whose largest part is [[alpha].sub.max], and let the entries in column i for 1 [less than or equal to] i [less than or equal to] [[alpha].sub.max] read from top to bottom be some word [w.sup.i].
From the above lemmas it follows that if we can reach an SRCT [[tau].sub.2] starting from [[tau].sub.1] via a sequence of flips, where [[tau].sub.2] [not equal to] [[tau].sub.1], then there does not exist a sequence of flips that takes [[tau].sub.2] to [[tau].sub.1].
Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the C-linear span of SRCTs that are greater than or equal to an SRCT [[tau].sub.i] under the total order [[??].sup.t.sub.[alpha]].
Theorem 6.2 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an [H.sub.n](0)-module whose quasisymmetric characteristic is the quasisymmetric Schur function [S.sub.[alpha]], where [alpha] is the shape of the SRCT [[tau].sub.1].
Definition 7.2 Let [tau] be an SRCT of shape [alpha] whose largest part is [[alpha].sub.max], and let the entries in column i for 1 [less than or equal to] i [less than or equal to] [[alpha].sub.max] read from top to bottom be some word [w.sup.i].
An SRCT [tau] of shape [alpha] is said to be a source tableau if it satisfies the condition that for every i [not member of] Des([tau]) and satisfying i [not equal to] n, we have that i + 1 lies to the immediate left of i.
EWS/PNET/SRCTs are rare & morphologically very similar, butdoesn't have specific antigens that could be demonstrated with immunocytochemistry or they lose them when poorly differentiated, cross-reactivity exists between some
SRCTs, thereforeit is difficult for cytopathologists to obtain experience for rendering reliable diagnoses.
