Let μ and ν = (ν 1, . . . , ν k ) be partitions such that μ is obtained from ν by adding m parts of sizer. Descouens and Morita proved algebraically that the modified Macdonald polynomials
satisfy the identity
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
This note gives a bijective proof of the formula for all r ≤ ν k .
Loehr, N. A., & Niese, E. (2012). A bijective proof of a factorization formula for specialized Macdonald polynomials. Annals of Combinatorics, 16(4), 815-828.