q,t-Catalan Local Chain Explorer

S(m,h) head & tail symmetry for local-chain decompositions of Def(k)

Head and tail symmetry calculator


m = 0 first-stable row calculator

For a partition size d, this computes the first-stable row h = d + 4, k = 2d + 2, and Sd = binom(h,2) - k. It then lists the rooted intervals μ1 + 1 ≤ A ≤ Sd - ℓ(μ) - 1 and the resulting area multiplicities. Endpoint Dyck words are shown directly for the common rows d = 4–9, and the page can compute them live for other positive values of d.


Local chain explorer

After computing symmetry data above, pick a local chain to see its elements.



            


            

                
Distinguished chain explorer

                
Iterated ν̃₂, ν̃₃, and a related ν̃₂ variant from the standard starting vectors. Each step is a Dyck vector along the distinguished chain; row i is paired with row N−i for the forced ζH symmetry on the all-1-coefficient families.

                

                    

                        

                        
                    

                    
                    
                    
                

                

                

                

            


            

                
Batch summary