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

%I A274644 #30 Mar 07 2025 02:56:12
%S A274644 1,6,71,1266,30206,902796,32420011,1359292626,65164480466,
%T A274644 3515569641156,210779736073446,13903319821066836,1000559812125494076,
%U A274644 78012524487061315416,6550837823204594551731,589404446176366002280146,56568586570039148217467786,5768723174387469795772704276,622900652040379217092492454866
%N A274644 Number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))].
%C A274644 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).
%H A274644 Michael Wallner, <a href="/A274644/b274644.txt">Table of n, a(n) for n = 1..100</a>
%H A274644 Cyril Banderier and Michael Wallner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/47.html">Young Tableaux with Periodic Walls: Counting with the Density Method</a>, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
%H A274644 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/p1.html">Problem 1</a>, Project P.
%H A274644 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/solutions/OneLevelGridPoset.pdf">Algorithmic Solution to Problem 1 (and linear extensions of general one-level grid-like posets)</a>, Project P.
%F A274644 From _Michael Wallner_, Feb 13 2024: (Start)
%F A274644 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.
%F A274644 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)
%p A274644 M := 20;
%p A274644 for k from 3 to 3+2*M do
%p A274644    bb[1,k] := 1;
%p A274644 end:
%p A274644 for n from 2 to M do
%p A274644 for k from 3 to 3+2*M-2*(n-1) do
%p A274644    bb[n,k] := sum(i*bb[n-1,i+2],i=1..k);
%p A274644 end;
%p A274644 end:
%p A274644 seq(bb[n,3],n=1..10);
%p A274644 N := 100:
%p A274644 f[1] := y-x;
%p A274644 for n from 1 to N-1 do
%p A274644    f[n+1] := (y-x)*int(int(subs(x=v,y=w,f[n]),w=v..y),v=0..x);
%p A274644 end:
%p A274644 for n from 1 to N do
%p A274644    aa[n] := factorial(3*n)*int(int(f[n],x=0..y),y=0..1);
%p A274644 end:
%p A274644 seq(aa[n],n=1..10);
%p A274644 # _Michael Wallner_, Feb 13 2024
%Y A274644 Cf. A000108, A274763.
%K A274644 nonn
%O A274644 1,2
%A A274644 _Ran Pan_, Jun 30 2016
%E A274644 All terms starting with a(13) corrected by _Michael Wallner_, Feb 13 2024