Bijecting hidden symmetries for skew staircase shapes

We present a bijection between the set SYT ( λ / μ ) of standard Young tableaux of staircase minus rectangle shape λ = δ k , μ = ( b a ) , and the set ShSYT ′ ( η ) of marked shifted standard Young tableaux of a certain shifted shape η = η ( k , a , b ) . Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for | SYT ( λ / μ ) | . This resolves an open problem by Morales, Pak and Panova (2019), and allows an efficient random sampling from SYT ( λ / μ ) . Other applications include a bijection for semistandard Young tableaux, and a bijective proof of Stembridge’s symmetry of LR–coefficients of the staircase shape. We also extend these results to set-valued standard Young tableaux in the combinatorics of K -theory , leading to new proofs of results by Lewis and Marberg (2019) and Abney-McPeek, An and Ng (2020).