Per W. Alexandersson - cp's OEIS frontend

A359039 Number of Wachs permutations of size n.

1, 1, 2, 4, 8, 24, 48, 192, 384, 1920, 3840, 23040, 46080, 322560, 645120, 5160960, 10321920, 92897280, 185794560, 1857945600, 3715891200, 40874803200, 81749606400, 980995276800, 1961990553600, 25505877196800, 51011754393600, 714164561510400, 1428329123020800

Offset: 0

Per W. Alexandersson, Dec 13 2022

nonn

A Wachs permutation pi is a permutation of [n] such that |pi^{-1}(i) - pi^{-1}(i*)| <= 1, for all 1 <= i <= n-1, where i* is defined as i-1 if i is even, i+1 if i is odd and i+1 <= n, and n otherwise.

			For n=4, a(n)=8, since we have the 8 Wachs permutations 1234, 1243, 2134, 2143, 3412, 3421, 4312, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..806
Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.

Cf. A016116, A081123.

A359039 := proc(n)
    local m ;
    m := floor(n/2) ;
    if type(n,'even') then
        m!*2^m ;
    else
        (m+1)!*2^m ;
    end if;
end proc: # R. J. Mathar, Jul 17 2023
# second Maple program:
a:= n-> ceil(n/2)!*2^floor(n/2):
seq(a(n), n=0..28);  # Alois P. Heinz, Dec 21 2023

a[n_]:=If[EvenQ[n], (n/2)! 2^(n/2), ((n + 1)/2)!*2^((n - 1)/2)]

If n=2m, then a(n) = m!*2^m, if n=2m+1, then a(n) = (m+1)!*2^m.
a(n) = A081123(n+1)*A016116(n). - Alois P. Heinz, Jan 23 2023
Sum_{n>=0} 1/a(n) = 3*sqrt(e) - 2. - Amiram Eldar, Jan 25 2023
D-finite with recurrence a(n) +2*a(n-1) +(-n-1)*a(n-2) +2*(-n+1)*a(n-3)=0. - R. J. Mathar, Jul 17 2023

A356185 The difference between number of even and number of odd Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 0, 1, 0, 3, 2, 9, 8, 23, 22, 53, 52, 115, 114, 241, 240, 495, 494, 1005, 1004, 2027, 2026, 4073, 4072, 8167, 8166, 16357, 16356, 32739, 32738, 65505, 65504, 131039, 131038, 262109, 262108, 524251, 524250, 1048537, 1048536, 2097111, 2097110, 4194261, 4194260

Offset: 0

Per W. Alexandersson, Jul 28 2022

A permutation is Grassmann if it has at most one descent. A closed-form formula was proved by J. B. Gil and J. A. Tomasko.

			For n=3, 123, 231, 312 are even Grassmann permutations, and 132, 213 are the odd ones. Hence a(3) = 1.

Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, Enum. Combin. Appl. 2 (2022), no. 4, Article #S4PP6.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Cf. A000325, A001710, A032085, A060546, A122746, A233411.

Bisections give: A005803 (even part), A183155 (odd part).

Table[2^Floor[1 + (n - 1)/2] - n, {n, 1, 80}]

a(n) = 2^(1+floor((n-1)/2))-n.
From Alois P. Heinz, Jul 28 2022: (Start)
G.f.: -(4*x^3-3*x^2-x+1)/((2*x^2-1)*(x-1)^2).
a(n) = A000325(n) - A233411(n) = A060546(n) - n = 2^ceiling(n/2) - n.
a(n) = A000325(n) - 2*A032085(n) = A000325(n) - 2*A122746(n-2) for n>=2. (End)

A355089 Number of parity-alternating permutations of [n] avoiding the pattern 123.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 10, 11, 37, 44, 146, 185, 603, 808, 2576, 3635, 11294, 16736, 50545, 78466, 230012, 373203, 1061236, 1795611, 4953447, 8721086, 23350320, 42691298, 111013825, 210379132, 531720722

Offset: 0

Per W. Alexandersson, Jun 18 2022

nonn

A permutation is parity-alternating if it sends odd integers to odd integers, and even integers to even integers. It avoids 123 if there is no subsequence a..b..c with a < b < c. The values are computed by Michael Albert, see MathOverflow link.

The odd-indexed entries agree with the odd-indexed entries in A354208. A bijection is given by reversing the permutation.

			For n=4, the three permutations are 3412, 3214, 1432.
For n=5, we have 54321, 52143, 32541.
For n=6, we have 563412, 563214, 543612, 543216, 561432, 541632, 365412, 365214, 321654, 165432.

Per Alexandersson, Samuel Asefa Fufa, Frether Getachew and Dun Qiu, Pattern-avoidance and Fuss-Catalan numbers, arXiv:2201.08168 [math.CO], 2022. See also J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
MathOverflow, 321-avoiding and parity-alternating permutations, Jun 06 2022.

Cf. A000108 (123-avoiding permutations), A010551 (parity-alternating permutations), A354208 (parity-alternating 321-avoiding permutations).

A354208 Number of parity-alternating permutations of [n] avoiding the pattern 321.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 11, 22, 44, 89, 185, 382, 808, 1702, 3635, 7779, 16736, 36229, 78466, 171238, 373203, 819186, 1795611, 3958662, 8721086, 19294525, 42691298, 94733886, 210379132, 468084856, 1042703207, 2325575076, 5193931583, 11609749877, 25986720374, 58203955771

Offset: 0

Per W. Alexandersson, Jun 06 2022

nonn

A permutation is parity-alternating if it sends odd integers to odd integers, and even integers to even integers. It avoids 321 if there is no subsequence a..b..c with a > b > c. The values are computed by Michael Albert, see MathOverflow link.

			For n=4, the two permutations are 1234, 3412.
For n=5, we have 12345, 34125, 14523.
For n=6, we have 123456, 341256, 145236, 125634, 561234, 345612.

Peter J. Taylor, Table of n, a(n) for n = 0..50
Per Alexandersson, Samuel Asefa Fufa, Frether Getachew and Dun Qiu, Pattern-avoidance and Fuss-Catalan numbers, arXiv:2201.08168 [math.CO], 2022. See also J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
MathOverflow, 321-avoiding and parity-alternating permutations, Jun 06 2022.
Nathan Williams, Oberwolfach Problem Session: Enumerative Combinatorics 2022, Univ. Texas Dallas (2023). See example 3.4, where this sequence is misidentified by typographical error.

Cf. A000108 (321-avoiding permutations), A010551 (parity-alternating permutations).

Offset corrected and terms a(30) and beyond from Peter J. Taylor, Jun 10 2022

A335340 North-East paths from (0,0) to (n,n) with k cyclic descents.

Original entry on oeis.org

2, 4, 2, 6, 12, 2, 8, 36, 24, 2, 10, 80, 120, 40, 2, 12, 150, 400, 300, 60, 2, 14, 252, 1050, 1400, 630, 84, 2, 16, 392, 2352, 4900, 3920, 1176, 112, 2, 18, 576, 4704, 14112, 17640, 9408, 2016, 144, 2, 20, 810, 8640, 35280, 63504, 52920, 20160, 3240, 180, 2

Offset: 1

Per W. Alexandersson, Jun 02 2020

A North-East path is a path from (0,0) to (n,n) using steps (1,0) and (0,1). A cyclic descent is a North step followed by an East step, where the last and first step is a cyclic descent if the path ends with a North step and starts with an East step.
The sum of the entries in row n is equal to binomial(2n,n).
I conjecture that the polynomial Sum_{k=1...n} T(n,k) t^k is real-rooted for all n.

			The table starts as
2,
4, 2
6, 12, 2
8, 36, 24, 2
10, 80, 120, 40, 2
12, 150, 400, 300, 60, 2

Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.

Cf. A103371.

T[n_, k_] = 2 Binomial[n, k] Binomial[n - 1, k - 1];

T(n,k) = 2*binomial(n,k)*binomial(n-1,k-1).

T(n,k) = 2 * A103371(n-1,k-1). - Alois P. Heinz, Jun 02 2020

A332389 Number A(n,w) of circular Dyck paths with n entries, and width at most w.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 18, 10, 5, 1, 32, 47, 28, 13, 6, 1, 64, 123, 82, 38, 16, 7, 1, 128, 322, 244, 117, 48, 19, 8, 1, 256, 843, 730, 370, 152, 58, 22, 9, 1, 512, 2207, 2188, 1186, 496, 187, 68, 25, 10, 1, 1024, 5778, 6562, 3827, 1648, 622, 222, 78, 28, 11

Offset: 1

Per W. Alexandersson, Feb 10 2020

A(n,w) is the number of circular Dyck paths of size n, and width at most w.
This is also the number of circular area lists, a_1, a_2, ..., a_n such that 0 <= a_i <= w-1, and a_{i+1} < a_i + 1, for all 1 <= i <= n, and the index i is taken modulo n.
The values of w are given by the row index.
A(n,w) is given by summing binomial(2*n - 1, n - 1 - (w+2) k) - binomial(2*n - 1, n + j + (w+2)*k) over k=1..w and k over all integers.

			The table begins as
1,    2,    3,    4,    5, ...
1,    4,    7,    10,   13, ...
1,    8,    18,   28,   38, ...
1,    16,   47,   82,   117, ...
1,    32,   123,  244,  370, ...
...
A(5,3)=123 and a few of the corresponding circular area lists are 00000, 10000,...,12210,...,12222, 22222.

Per Alexandersson, Svante Linusson and Samu Potka, The cyclic sieving phenomenon on circular Dyck paths, arXiv:1903.01327 [math.CO], 2019.
Per Alexandersson, Svante Linusson and Samu Potka, The cyclic sieving phenomenon on circular Dyck paths, Electronic Journal of Combinatorics 26, No.4 (2019).

A194460 is the diagonal.

CircularDyckPaths[n_, w_] := With[{d = w + 2},
   Sum[Binomial[2 n - 1, n - 1 - d s] -
     Binomial[2 n - 1, n + j + d s]
    , {j, w},
    {s, -2 (n + 2), 2 (n + 2)}]
   ];
Table[
CircularDyckPaths[n, w]
, {n, 1, 10}, {w, 1, 10}]

A(n,w) = Sum_{k=-2*(n+2)..2*(n+2)} Sum_{j=1..w} binomial(2n-1, n-1-(w+2)*k) - binomial(2*n-1, n + j + (w+2)*k).

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User: Per W. Alexandersson

A359039 Number of Wachs permutations of size n.

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A356185 The difference between number of even and number of odd Grassmannian permutations of size n.

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A355089 Number of parity-alternating permutations of [n] avoiding the pattern 123.

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A354208 Number of parity-alternating permutations of [n] avoiding the pattern 321.

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A335340 North-East paths from (0,0) to (n,n) with k cyclic descents.

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A332389 Number A(n,w) of circular Dyck paths with n entries, and width at most w.

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