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A274644
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Number of linear extensions of the one-level grid poset G[(1^n), (0^(n-1)), (0^(n-1))].
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6
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1, 6, 71, 1266, 30206, 902796, 32420011, 1359292626, 65164480466, 3515569641156, 210779736073446, 13903319821066836, 1000559812125494076, 78012524487061315416, 6550837823204594551731, 589404446176366002280146, 56568586570039148217467786, 5768723174387469795772704276, 622900652040379217092492454866
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) = b(n,3) in b(n,k) = Sum_{1 <= i <=k} i*b(n-1,i+2) for n>0 and k>=3 with initial conditions b(1,k)=1 for all k. - Michael Wallner, Feb 13 2024
a(n) = (3*n)!*int(int(f_{n}(x,y),x=0..y),y=0..1) where f_{n+1} = (y-x)*int(int(f_{n}(v,w)),w=v..y),v=0..x)) for n>=1 and f_{1}(x,y) = y-x (Derived using the density method; see [Banderier, Wallner 2021]). - Michael Wallner, Feb 13 2024
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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