Séminaire Lotharingien de Combinatoire, 95B.100 (2026), 12 pp.
Suchakree Chueluecha and Jennifer Morse
Skew-Linked Catalan Functions and Genocchi Numbers
Abstract.
The discovery that k-Schur functions satisfy a striking shift invariance property and arise as a subfamily of Catalan functions resolved long-standing mysteries in the theory. These Catalan functions are symmetric functions realized as GLℓ-equivariant Euler characteristics on the flag variety. This suggests that the phenomena surrounding k-Schur functions might reflect a more general mechanism inside the Catalan-function framework. Indeed, we identify a broad class of {\it steep} Catalan functions and show that all these elements satisfy shift invariance. Moreover, the family of steep Catalan functions beautifully partitions into subclasses indexed by distinguished skew diagrams called skew-linked diagrams.
We further isolate a notable subclass of steep Catalan functions that includes all k- and (k-1)-Schur functions and prove that they satisfy a powerful factorization property, enabling their construction from a Genocchi-number-sized subfamily. We investigate the combinatorics of Genocchi numbers and root ideals, establishing a new bijection between a natural family of root ideals and the Genocchi numbers.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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