scholarly journals Some Enumerations on Non-Decreasing Dyck Paths

10.37236/3941 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Rigoberto Flórez ◽  
Leandro Junes
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Maximum Element ◽  
Dyck Paths ◽  
Several Variables ◽  
Formal Power

We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all non-decreasing Dyck paths of length $2n.$ We also show that an indexed sum on pyramid weights depends only on the size and maximum element of the indexing set.

Substitution Groups of Formal Power Series

1954 ◽  
Vol 6 ◽  
pp. 325-340 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Point Of View ◽  
Group Operation ◽  
Several Variables ◽  
Special Cases ◽  
Image Position ◽  
Formal Power ◽  
Complex Coefficients

In this paper we are concerned with the group of formal power series of the form,the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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