OFFSET
0,3
COMMENTS
Dumont permutations of the fourth kind are permutations of even length where all deficiencies (drops) are even values at even positions.
REFERENCES
O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, Ph.D. thesis, Howard University, 2019.
LINKS
A. Burstein and O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
A. Burstein, M. Josuat-Vergès, and W. Stromquist, New Dumont permutations, Pure Math. Appl. (Pu.M.A.) 21 (2010), no. 2, 177-206.
FORMULA
Let F_k(x) be the truncation of the g.f. of A048990 to a polynomial of degree k. Let G_k(x) be the truncation of the g.f. of A024492 to a polynomial of degree k. Let G_{-1}(x) = 0. For k>=0, define A_k(x) recursively as follows: A_k(x) = F_k(x)/((1-x*G_{k-1}(x))^2-x*F_k(x)/(1-x*G_k(x)-x*F_k(x)^2/(1-x*A_{k+1}(x)))). Then A_0(x) is the g.f. of this sequence.
EXAMPLE
For n=2, a(2)=3 counts the permutations 1234, 1342, 1432.
PROG
(PARI) seq(n)={my(h=sqrt(1-16*x + O(x*x^n)), F=sqrt((1-h)/(8*x)), G=(1-sqrt((1+h)/2))/(2*x), A=O(1)); forstep(k=n\3, 0, -1, my(f=Pol(F + O(x*x^k))); A = f/((1 - x*Pol(G + O(x^k)))^2 - x*f/(1 - x*Pol(G + O(x*x^k)) - x*f^2/(1 - x*A))) ); Vec(A + O(x*x^n))} \\ Andrew Howroyd, Apr 29 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Alexander Burstein and Opel Jones, Apr 29 2021
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Apr 29 2021
STATUS
approved