OFFSET
1,2
COMMENTS
The definition of a one-level grid poset can be found in the Pan links. The number of linear extensions of the one-level grid poset G[(0^n), (0^(n-1)), (0^(n-1))] is given by Catalan number A000108(n) and the number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))] is given by A274644(n).
LINKS
Cyril Banderier and Michael Wallner, Young Tableaux with Periodic Walls: Counting with the Density Method, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
Ran Pan, Problem 1, Project P.
FORMULA
a(n) = (4*n-1)!*int(int(f_{n}(x,y),x=0..y),y=0..1) where f_{n+1} = (y-x)*int(int((x-u)*f_{n}(u,v)),v=u..y),u=0..x)) for n>=1 and f_{1}(x,y) = y-x (Derived using the density method; see [Banderier, Wallner 2021]). - Michael Wallner, Feb 14 2024
MAPLE
N := 100;
ff[1] := y-x;
for n from 1 to N-1 do
ff[n+1] := simplify((y-x)*int(int((x-u)*subs(x=u, y=v, ff[n]), v=u..y), u=0..x));
end:
for n from 1 to N do
a[n] := factorial(4*n-1)*int(int(ff[n], x=0..y), y=0..1);
end:
seq(a[n], n=1..10);
# Michael Wallner, Feb 14 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Ran Pan, Jul 05 2016
EXTENSIONS
Corrected and extended by Michael Wallner, Feb 14 2024
STATUS
approved