
Permutations and the Rest. Algebraic Combinatorics of Permutations. 285
of a Standard Young Tableau Z if i appears in a row in Z that is strictly
above the row in which i + 1 appears in Z.
THEOREM 7.15
Let π ∈ S
n
,andleti ∈ [n − 1].Theni is a descent of π if and only if i is a
descent of Q(π).
PROOF First let us assume that i ∈ D(p), that is, π
i
>π
i+1
. We need
to show that the insertion of π
i+1
results in the addition of a new box to the
P -tableaux that is below the box resulting from the insertion of π
i
.
As π
i
>π
i+1
, we know that π
i+1
gets inserted to the first row of P (π)
weakly on the left of π
i
. If the insertion of π
i
ended in the first row, then we
are ...