A274763 - OEIS

A274763

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

1, 10, 215, 7200, 328090, 18914190, 1318595475, 107813147200, 10112867995550, 1070215246700100, 126122386636230950, 16378717184245443000, 2323753119238888045500, 357594668486650175355750, 59323244552378848484536875, 10553747415214416889115286000, 2004246729406751177924041663750, 404685181230584369889138573637500, 86569650968075614116679243211951250, 19558042902565983702641321883519060000

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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

Table of n, a(n) for n=1..20.

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.

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