Bijective recurrences concerning two Schröder triangles

by   Shishuo Fu, et al.

Let r(n,k) (resp. s(n,k)) be the number of Schröder paths (resp. little Schröder paths) of length 2n with k hills, and set r(0,0)=s(0,0)=1. We bijectively establish the following recurrence relations: r(n,0) =∑_j=0^n-12^jr(n-1,j), r(n,k) =r(n-1,k-1)+∑_j=k^n-12^j-kr(n-1,j), 1< k< n, s(n,0) =∑_j=1^n-12·3^j-1s(n-1,j), s(n,k) =s(n-1,k-1)+∑_j=k+1^n-12·3^j-k-1s(n-1,j), 1< k< n. The infinite lower triangular matrices [r(n,k)]_n,k> 0 and [s(n,k)]_n,k> 0, whose row sums produce the large and little Schröder numbers respectively, are two Riordan arrays of Bell type. Hence the above recurrences can also be deduced from their A- and Z-sequences characterizations. On the other hand, it is well-known that the large Schröder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run, whose distribution on separable permutations is shown to be given by [r(n,k)]_n,k> 0 as well.


page 1

page 2

page 3

page 4


Refined Wilf-equivalences by Comtet statistics

We launch a systematic study of the refined Wilf-equivalences by the sta...

Bijective proofs for Eulerian numbers in types B and D

Let ⟨n k⟩, ⟨B_n k⟩, and ⟨D_n k⟩ be the Eulerian numbers in the types A, ...

On digital sequences associated with Pascal's triangle

We consider the sequence of integers whose nth term has base-p expansion...

On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond

We consider the following variant of the Mortality Problem: given k× k m...

q-Stirling numbers arising from vincular patterns

The distribution of certain Mahonian statistic (called BAST) introduced ...

A Bijection Between Weighted Dyck Paths and 1234-avoiding Up-Down Permutations

Three-dimensional Catalan numbers are a variant of the classical (bidime...

Please sign up or login with your details

Forgot password? Click here to reset