A024492 -id:A024492 - cp's OEIS frontend

A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

1, 1, 3, 10, 39, 174, 872, 4805, 28474, 178099, 1160173, 7803860, 53924841, 381640934, 2761331130, 20400560942, 153738854242, 1180631743440, 9229687049249, 73372263658451, 592476077260123, 4854377724124700, 40315729803287046, 339065862485375334, 2885324166565733641

Offset: 0

Alexander Burstein and Opel Jones, Apr 29 2021

Dumont permutations of the fourth kind are permutations of even length where all deficiencies (drops) are even values at even positions.

			For n=2, a(2)=3 counts the permutations 1234, 1342, 1432.

O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, Ph.D. thesis, Howard University, 2019.

A. Burstein and O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
A. Burstein, M. Josuat-Vergès, and W. Stromquist, New Dumont permutations, Pure Math. Appl. (Pu.M.A.) 21 (2010), no. 2, 177-206.

Cf. A000108 (permutations avoiding 312), A024492, A048990, A110501 (length 2n Dumont permutations of 4th kind).

seq(n)={my(h=sqrt(1-16*x + O(x*x^n)), F=sqrt((1-h)/(8*x)), G=(1-sqrt((1+h)/2))/(2*x), A=O(1)); forstep(k=n\3, 0, -1, my(f=Pol(F + O(x*x^k))); A = f/((1 - x*Pol(G + O(x^k)))^2 - x*f/(1 - x*Pol(G + O(x*x^k)) - x*f^2/(1 - x*A))) ); Vec(A + O(x*x^n))} \\ Andrew Howroyd, Apr 29 2021

Let F_k(x) be the truncation of the g.f. of A048990 to a polynomial of degree k. Let G_k(x) be the truncation of the g.f. of A024492 to a polynomial of degree k. Let G_{-1}(x) = 0. For k>=0, define A_k(x) recursively as follows: A_k(x) = F_k(x)/((1-x*G_{k-1}(x))^2-x*F_k(x)/(1-x*G_k(x)-x*F_k(x)^2/(1-x*A_{k+1}(x)))). Then A_0(x) is the g.f. of this sequence.

Terms a(12) and beyond from Andrew Howroyd, Apr 29 2021

A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2k,2k).

Original entry on oeis.org

1, 2, 6, 14, 52, 140, 558, 1598, 6604, 19588, 82780, 251212, 1077992, 3324760, 14427422, 45039422, 197122524, 621205076, 2737289748, 8691699524, 38510822360, 123045322024, 547682980716, 1759017606220, 7859796084984, 25355507376808, 113670929821304

Offset: 0

John Tyler Rascoe, Oct 08 2024

			The path NENNEENE does not cross y = x, so it is counted under a(4) = 52.
The path NENNENNEEEEN crosses y = x at points (1,1) and (5,5), so it is counted under a(6) = 558.

Cf. A000108, A000984, A024492, A048990, A268400, A268429.

C(x) = {(1-sqrt(1-4*x))/(2*x)}
A(x) ={C(4*x)*C((x)*C(4*x))}
B(x) = {sqrt(C(4*x))}
D(x) = {1/sqrt(1-4*x)}
E_x(N) = {my(x='x+O('x^N));  Vec(D(x)-2*((C(x)-1)*((x*A(x^2))^2-B(x^2)^2+3*B(x^2)-2))/((2-B(x^2))*(2-C(x))))}
E_x(30)

G.f. D(x) - 2*((C(x) - 1)*((x*A(x^2))^2 - B(x^2)^2 + 3*B(x^2) - 2))/((2 - B(x^2))*(2 - C(x))), where A(x), B(x), C(x), and D(x) are the g.f.s for A024492, A048990, A000108, and A000984.

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A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

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A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2k,2k).

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cp's OEIS Frontend

A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

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Keywords

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Programs

PARI

Formula

Extensions

A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2*k,2*k).

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A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2k,2k).