Abstract
Let μ and ν = (ν 1, . . . , ν k ) be partitions such that μ is obtained from ν by adding m parts of size r. Descouens and Morita proved algebraically that the modified Macdonald polynomials \({{\tilde{H}_\mu}(X; q, t)}\) satisfy the identity \({{\tilde{H}_\mu} = \tilde{H}_\nu \tilde{H}_{(r^m)}}\) when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ ν k and \({r \in \{1, 2\}}\) . This note gives a bijective proof of the formula for all r ≤ ν k .
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Loehr, N.A., Niese, E. A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials. Ann. Comb. 16, 815–828 (2012). https://doi.org/10.1007/s00026-012-0162-5
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DOI: https://doi.org/10.1007/s00026-012-0162-5