A274644 Number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))].

1, 6, 71, 1266, 30206, 902796, 32420011, 1359292626, 65164480466, 3515569641156, 210779736073446, 13903319821066836, 1000559812125494076, 78012524487061315416, 6550837823204594551731, 589404446176366002280146, 56568586570039148217467786, 5768723174387469795772704276, 622900652040379217092492454866

Offset: 1

Author

Ran Pan, Jun 30 2016

Keywords

• nonn

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).

Links

• Michael Wallner, Table of n, a(n) for n = 1..100
• 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.
• Ran Pan, Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets), Project P.

Crossrefs

Cf. A000108, A274763.

Programs

Maple

M := 20;
for k from 3 to 3+2*M do
   bb[1,k] := 1;
end:
for n from 2 to M do
for k from 3 to 3+2*M-2*(n-1) do
   bb[n,k] := sum(i*bb[n-1,i+2],i=1..k);
end;
end:
seq(bb[n,3],n=1..10);
N := 100:
f[1] := y-x;
for n from 1 to N-1 do
   f[n+1] := (y-x)*int(int(subs(x=v,y=w,f[n]),w=v..y),v=0..x);
end:
for n from 1 to N do
   aa[n] := factorial(3*n)*int(int(f[n],x=0..y),y=0..1);
end:
seq(aa[n],n=1..10);
# Michael Wallner, Feb 13 2024

Formula

From Michael Wallner, Feb 13 2024: (Start)

a(n) = b(n,3) in b(n,k) = Sum_{i=1..k} i*b(n-1,i+2) for n>0 and k>=3 with initial conditions b(1,k) = 1 for all k.

a(n) = (3*n)!*Integral_{y=0..1} Integral_{x=0..y} f_{n}(x,y) dx dy where f_{n+1}(x,y) = (y-x)*Integral_{v=0..x} Integral_{w=v..y} f_{n}(v,w) dw dv for n>=1 and f_{1}(x,y) = y-x (Derived using the density method; see [Banderier, Wallner 2021]). (End)

Extensions

All terms starting with a(13) corrected by Michael Wallner, Feb 13 2024