Refining Blecher and Knopfmacher's Integer Partition Fixed Points

Recently, Blecher and Knopfmacher explored the notion of fixed points in integer partitions. Here, we distinguish partitions with a fixed point by which value is fixed and analyze the resulting triangle of integers. In particular, we confirm various identities for diagonal sums, row sums, and antidiagonal sums (which are finite for this triangle) and establish a four-term recurrence for triangle entries analogous to Pascal’s lemma for the triangle of binomial coefficients. The partition statistics crank and mex arise. All proofs are combinatorial.