Define matrix indexed by partitions

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I am not sure about the notation $\lambda'$ so let me assume that it is the conjugate.

First, to build a matrix, you have to index the partitions, which is easy with the enumerate function:

sage: n = 6
sage: P = Partitions(n)
sage: m = matrix(ZZ, P.cardinality())
sage: list(enumerate(P))
[(0, [6]),
 (1, [5, 1]),
 (2, [4, 2]),
 (3, [4, 1, 1]),
 (4, [3, 3]),
 (5, [3, 2, 1]),
 (6, [3, 1, 1, 1]),
 (7, [2, 2, 2]),
 (8, [2, 2, 1, 1]),
 (9, [2, 1, 1, 1, 1]),
 (10, [1, 1, 1, 1, 1, 1])]

In the other way, given a partition, you need to be able to find its index, which is doable with this dictionary:

sage: part_to_int = {p:i for i,p in enumerate(P)}

Now, you can construct your matrix:

sage: for i,p in enumerate(P):
....:     j = part_to_int[p.conjugate()]
....:     m[i,j] = 1

You have:

sage: m
[0 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0 0]

sage: m^2
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1]

If $\lambda'$ does not denote the conjugate, you can adapt by replacing p.conjugate() with your favorite function.