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A274763 Number of linear extensions of the one-level grid poset G[(1^n), (1^(n-1)), (0^(n-1))].

%I A274763 #21 Mar 07 2025 03:40:47
%S A274763 1,10,215,7200,328090,18914190,1318595475,107813147200,10112867995550,
%T A274763 1070215246700100,126122386636230950,16378717184245443000,
%U A274763 2323753119238888045500,357594668486650175355750,59323244552378848484536875,10553747415214416889115286000,2004246729406751177924041663750,404685181230584369889138573637500,86569650968075614116679243211951250,19558042902565983702641321883519060000
%N A274763 Number of linear extensions of the one-level grid poset G[(1^n), (1^(n-1)), (0^(n-1))].
%C A274763 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).
%H A274763 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 A274763 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/p1.html">Problem 1</a>, Project P.
%H A274763 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 A274763 a(n) = (4*n-1)!*Integral_{y=0..1} Integral_{x=0..y} f_{n}(x,y) dx dy where f_{n+1}(x,y) = (y-x)*Integral_{u=0..x} Integral_{v=u..y} (x-u)*f_{n}(u,v) dv du for n>=1 and f_{1}(x,y) = y-x (Derived using the density method; see [Banderier, Wallner 2021]). - _Michael Wallner_, Feb 14 2024
%p A274763 N := 100;
%p A274763 ff[1] := y-x;
%p A274763 for n from 1 to N-1 do
%p A274763    ff[n+1] := simplify((y-x)*int(int((x-u)*subs(x=u,y=v,ff[n]),v=u..y),u=0..x));
%p A274763 end:
%p A274763 for n from 1 to N do
%p A274763    a[n] := factorial(4*n-1)*int(int(ff[n],x=0..y),y=0..1);
%p A274763 end:
%p A274763 seq(a[n],n=1..10);
%p A274763 # _Michael Wallner_, Feb 14 2024
%Y A274763 Cf. A000108, A274644.
%K A274763 nonn
%O A274763 1,2
%A A274763 _Ran Pan_, Jul 05 2016
%E A274763 Corrected and extended by _Michael Wallner_, Feb 14 2024