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Séminaire Lotharingien de Combinatoire, B36h (1996), 24pp.

# Rudolf Winkel

# A Combinatorial Bijection Between
Standard Young Tableaux and
Reduced Words of Grassmannian Permutations

**Abstract.**
For every partition *\lambda* we construct
a very simple combinatorial bijection between the set of standard
Young tableaux of shape *\lambda* and the set of reduced words for the
Grassmannian permutation *\pi*(*\lambda*) associated to *\lambda*. The
basic tools in setting up this bijection are partial orders on the
respective sets. These partial orders are interesting in their own right,
and we give some first results about them: (1) the poset of
standard tableaux for an arbitrary shape *D* is isomorphic to an order ideal
in left weak Bruhat order, (2) for hook shapes the Poincaré polynomial
is the *q*-binomial coefficient, (3) for general Ferrer shapes a
recursion formula for the Poincaré polynomials is given, (4)
the poset of reduced words for a Grassmannian permutation is
anti-isomorphic to the poset of reduced words for its ``conjugate'' and
inverse permutation,
(5) for the Grassmannian and dominant permutation associated to a hook shape
the respective posets of reduced words are isomorphic.

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