With epsilon invertible and delta arbitrary, Rook-Brauer homology is isomorphic to symmetric group homology in all degrees and Motzkin homology vanishes in positive degrees.
3, 1437-1499, available athttps://msp.org/gt/2024/28-3/gt-v28-n3-p11-s.pdf
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Homology of Rook-Brauer Algebras and Motzkin Algebras
With epsilon invertible and delta arbitrary, Rook-Brauer homology is isomorphic to symmetric group homology in all degrees and Motzkin homology vanishes in positive degrees.