  The decomposition matrix of a finite group in prime characteristic p records
the multiplicities of its p-modular irreducible representations as composition
factors of the reductions modulo p of its irreducible representations in
characteristic zero. The main theorem of this paper gives a combinatorial
description of certain columns of the decomposition matrices of symmetric
groups in odd prime characteristic. The result applies to blocks of arbitrarily
high weight. It is obtained by studying the p-local structure of certain twists
of the permutation module given by the action of the symmetric group of even
degree 2m on the collection of set partitions of a set of size 2m into m sets
each of size two. In particular, the vertices of the indecomposable summands of
all such modules are characterized; these summands form a new family of
indecomposable p-permutation modules for the symmetric group. As a further
corollary it is shown that for every natural number w there is a diagonal
Cartan number in a block of the symmetric group of weight w equal to w+1.
