The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

Guo-Niu Han

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Hooks and powers of parts in partitions.

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Enumeration of sequences constrained by the ratio of consecutive parts.

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Overpartition pairs

Jeremy Lovejoy (2006)

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An overpartition pair is a combinatorial object associated with the $q$ -Gauss identity and the $_{1} ψ_{1}$ summation. In this paper, we prove identities for certain restricted overpartition pairs using Andrews’ theory of recurrences for well-poised basic hypergeometric series and the theory of Bailey chains.

The major counting of nonintersecting lattice paths and generating functions for tableaux. (Summary).

Krattenthaler, C. (1992)

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Advanced determinant calculus.

Krattenthaler, Christian (1999)

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Adding $\pm 1$ to the argument of a Hall-Littlewood polynomial.

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The higher transvectants are redundant

Abdelmalek Abdesselam, Jaydeep Chipalkatti (2009)

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Let $A, B$ denote generic binary forms, and let $𝔲_{r} = {(A, B)}_{r}$ denote their $r$ -th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the ${𝔲_{r}}$ . As a consequence, we show that each of the higher transvectants ${𝔲_{r} : r \geq 2}$ is redundant in the sense that it can be completely recovered from $𝔲_{0}$ and $𝔲_{1}$ . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S L_{2}$ -representations,...