A081123 -id:A081123 - cp's OEIS frontend

A231777 Number T(n,k) of permutations of [n] with exactly k ascents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

1, 1, 1, 1, 4, 2, 8, 14, 2, 54, 60, 6, 162, 402, 150, 6, 1536, 2712, 768, 24, 6144, 19704, 12744, 1704, 24, 75000, 183120, 94320, 10320, 120, 375000, 1473720, 1392720, 365520, 21720, 120, 5598720, 17522640, 13631040, 3011040, 152640, 720, 33592320, 156250800

Offset: 0

Alois P. Heinz, Nov 13 2013

			T(4,0) = 8: 1324, 2413, 2431, 3241, 4132, 4213, 4231, 4321.
T(4,1) = 14: 1243, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 3124, 3142, 3214, 3421, 4123, 4312.
T(4,2) = 2: 1234, 3412.
T(5,2) = 6: 12345, 12534, 34125, 34512, 51234, 53412.
T(6,3) = 6: 123456, 125634, 341256, 345612, 561234, 563412.
Triangle T(n,k) begins:
:  0 :      1;
:  1 :      1;
:  2 :      1,       1;
:  3 :      4,       2;
:  4 :      8,      14,       2;
:  5 :     54,      60,       6;
:  6 :    162,     402,     150,      6;
:  7 :   1536,    2712,     768,     24;
:  8 :   6144,   19704,   12744,   1704,    24;
:  9 :  75000,  183120,   94320,  10320,   120;
: 10 : 375000, 1473720, 1392720, 365520, 21720, 120;

Alois P. Heinz, Rows n = 0..23, flattened

Column k=0 gives: A231601.
Row sums and T(2n,n) give: A000142.
T(n,floor(n/2)) gives: A081123(n+1).
Cf. A004526.

A359039 Number of Wachs permutations of size n.

Original entry on oeis.org

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

A366109 a(n) = floor(n!(3floor(n/2)!ceiling(n/2)! + 3floor((n+2)/2)!ceiling((n-2)/2)! - 6floor(n/2)!*ceiling((n-2)/2)!)^(-1)).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 26, 46, 92, 168, 333, 616, 1225, 2288, 4558, 8580, 17107, 32413, 64664, 123170, 245832, 470288, 938943, 1802770, 3600207, 6933733, 13849778, 26744400, 53429368, 103411680, 206621384, 400720260, 800747232, 1555737480, 3109074130, 6050090200, 12091800773

Offset: 3

Stefano Spezia, Sep 29 2023

nonn

Gábor Czédli, Minimum-sized generating sets of the direct powers of the free distributive lattice on three generators and a Sperner theorem, arXiv:2309.13783 [math.CO], 2023. See formulas (3.6) at p. 4 and (4.15) at p. 8.

Cf. A000142, A004526, A081123, A218708, A366107, A366108.

a[n_]:=Floor[n!(3Floor[n/2]!Ceiling[n/2]! + 3Floor[(n+2)/2]!Ceiling[(n-2)/2]! - 6Floor[n/2]!Ceiling[(n-2)/2]!)^(-1)]; Array[a,37,3]

a(n)/A366107(n) ~ 7/6 (see Remark 3.4 at p. 5 in Czédli).

a(n) ~ c*2^n/sqrt(n), with c = 1/(3*sqrt(2*Pi)) = (2/3)*A218708.

A133922 a(n) is number of permutations (p(1),p(2),p(3),...,p(n)) of (1,2,3,...,n) such that p(k) is coprime to p(n+1-k) for k = all positive integers <= n.

Original entry on oeis.org

1, 2, 2, 16, 16, 192, 192, 6912, 4608, 230400, 230400, 11612160, 11612160, 1199923200, 588349440, 98594979840, 98594979840, 11076328488960, 11076328488960, 2102897147904000, 1076597725593600, 331238941183180800, 331238941183180800, 66325953940291584000, 56326771107377971200

Offset: 1

Leroy Quet, Jan 07 2008

For n = odd integer the middle term of all counted permutations must be 1.
From Robert Israel, Sep 12 2016: (Start)
Consider the graph with vertices [1,...,n] if n is even, [2,...,n] if n is odd, and edges joining coprime integers.
a(n) is A037223(n) times the number of perfect matchings in this graph.
If n is even, a(n) = A037223(n)*A009679(n/2).
If n is an odd prime, a(n) = a(n-1). (End)

			For n = 6, the permutation (3,2,1,6,4,5) is not counted because p(2)=2 is not coprime to p(5)=4. However, the permutation (3,6,1,4,5,2) is counted because GCD(3,2) = GCD(6,5) = GCD(1,4) = 1.

Robert Israel, Table of n, a(n) for n = 1..31

Cf. A009679, A081123, A037223.

M:= proc(A) option remember;
    local n,t,i,Ai,Ap,inds,isrt,As;
    n:= nops(A);
    if n = 0 then return 1 fi;
    t:= 0;
    for i in A[1] do
      inds:= [$2..i-1,$i+1..n];
      Ai:= subs([1=NULL,i=NULL,seq(inds[i]=i,i=1..n-2)],A[inds]);
      isrt:= sort([$1..n-2],(r,s) -> nops(Ai(r)) <= nops(Ai(s)),output=permutation);
      Ai:= subs([seq(isrt[i]=i,i=1..n-2)],Ai[isrt]);
      t:= t + procname(Ai);
    od;
    t;
end proc:
F:= proc(n) local A;
  if n::odd then
    if isprime(n) then return procname(n-1) fi;
    A:= [seq(select(t -> igcd(t+1,i+1)=1, [$1..i-1,$i+1..n-1]),i=1..n-1)];
  else
    A:= [seq(select(t -> igcd(t,i)=1,[$1..i-1,$i+1..n]),i=1..n)]
  fi;
  M(A)*floor(n/2)!*2^floor(n/2)
end proc;
seq(F(n),n=1..27); # Robert Israel, Sep 12 2016

a(6)-a(15) from Sean A. Irvine, May 17 2010

a(16)-a(25) from Robert Israel, Sep 12 2016

A249138 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 6, 5, 7, 1, 1, 6, 18, 8, 10, 1, 1, 24, 26, 46, 12, 14, 1, 1, 24, 96, 58, 86, 16, 18, 1, 1, 120, 154, 326, 118, 156, 21, 23, 1, 1, 120, 600, 444, 756, 198, 246, 26, 28, 1, 1, 720, 1044, 2556, 1152, 1692, 324, 384, 32, 34, 1, 1

Offset: 0

Clark Kimberling, Oct 22 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+2)/2)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A056952(n+2) for n >= 0.
(Column 1) is essentially A081123 (repeated factorials).

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
f(2,x) = (2 + x + x^2)/(1 + x), so that p(2,x) = 2 + x + x^2.
First 6 rows of the triangle of coefficients:
  1
  1    1
  2    1    1
  2    4    1    1
  6    5    7    1    1
  6    18   8    10   1   1

Clark Kimberling, Rows 0..100, flattened

Cf. A056952, A081123, A249128.

z = 15; p[x_, n_] := x + Floor[(n+1)/2]/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249138 array *)
Flatten[CoefficientList[u, x]] (* A249138 sequence *)

A355986 a(n) = Sum_{k=1..n} floor(n/k)!.

Original entry on oeis.org

1, 3, 8, 28, 125, 731, 5052, 40352, 362917, 3628935, 39916936, 479002360, 6227021561, 87178296283, 1307674373184, 20922789928484, 355687428136485, 6402373706091651, 121645100409195652, 2432902008180269688, 51090942171713074013, 1124000727777647602015

Offset: 1

Seiichi Manyama, Jul 22 2022

			a(6) = 6! + 3! + 2! + 1! + 1! + 1! = 731.

Cf. A007489, A060832, A081123.

a[n_] := Sum[Floor[n/k]!, {k, 1, n}]; Array[a, 22] (* Amiram Eldar, Jul 22 2022 *)

PARI
```
a(n) = sum(k=1, n, (n\k)!);
```

A361522 The aerated factorial numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 6, 0, 24, 0, 120, 0, 720, 0, 5040, 0, 40320, 0, 362880, 0, 3628800, 0, 39916800, 0, 479001600, 0, 6227020800, 0, 87178291200, 0, 1307674368000, 0, 20922789888000, 0, 355687428096000, 0, 6402373705728000, 0, 121645100408832000, 0, 2432902008176640000

Offset: 0

Peter Luschny, Mar 14 2023

nonn

An aerated version of A000142, which is the main entry for this sequence.

Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
Eric Weisstein's World of Mathematics, Error function erf.

Cf. A000142, A081124, A084261, A081123, A161556.

egf := (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1:
ser := series(egf, z, 42): seq(n!*coeff(ser, z, n), n = 0..40);

a[n_] := If[OddQ[n], 0, (n/2)!]; Array[a, 41, 0] (* Amiram Eldar, Mar 14 2023 *)

a(n) = n! * [z^n] (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1.

a(n) = n! * [z^n] 1 + 2*u*exp(u)*hypergeom([1/2], [3/2], -u), where u = (z/2)^2.

A373829 Number of inefficient arrangements in A373182, where inefficient means that the maximum number of persons that a seating arrangement can hold is not achieved.

Original entry on oeis.org

0, 0, 1, 0, 6, 2, 36, 24, 246, 240, 1920, 2424, 16920, 25920, 166440, 297360, 1809360, 3669840, 21551040, 48666240, 279180720, 691649280, 3908580480, 10501787520, 58813776000, 169809696000, 946627274880, 2914924320000, 16228733875200, 52963370208000

Offset: 1

Enrique Navarrete, Jun 19 2024

nonn

The maximum number of persons that can be seated in the arrangements in A373182 in n seats is ceiling(n/2).
The seatings here are maximal in the sense that no additional person can be seated without breaking the condition in A373182, but maximum seatings are excluded.
The ratio a(n)/A373182(n) -> 1 as n -> infinity (at a much slower initial rate for even n).

			a(5)=6 are the following seatings, where _ denotes an empty seat. Seatings of 3 people are the maximum for n=5 and those are not included.
    1 _ _ 2 _
    _ 1 _ 2 _
    _ 1 _ _ 2
    _ 2 _ 1 _
    2 _ _ 1 _
    _ 2 _ _ 1.
For n=9 seats the maximum number of persons that can be seated is 5, hence examples of inefficient arrangements are:
    3 _ 2 _ 1 _ _ 4 _
    _ 3 _ _ 1 _ _ 2 _.

Cf. A081123, A373182.

a(n) = A373182(n) - (ceiling((n+1)/2))!.

cp's OEIS Frontend

A231777 Number T(n,k) of permutations of [n] with exactly k ascents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

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A359039 Number of Wachs permutations of size n.

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A366109 a(n) = floor(n!*(3*floor(n/2)!*ceiling(n/2)! + 3*floor((n+2)/2)!*ceiling((n-2)/2)! - 6*floor(n/2)!*ceiling((n-2)/2)!)^(-1)).

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A133922 a(n) is number of permutations (p(1),p(2),p(3),...,p(n)) of (1,2,3,...,n) such that p(k) is coprime to p(n+1-k) for k = all positive integers <= n.

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A249138 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

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Mathematica

A355986 a(n) = Sum_{k=1..n} floor(n/k)!.

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PARI

A361522 The aerated factorial numbers.

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A373829 Number of inefficient arrangements in A373182, where inefficient means that the maximum number of persons that a seating arrangement can hold is not achieved.

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A366109 a(n) = floor(n!(3floor(n/2)!ceiling(n/2)! + 3floor((n+2)/2)!ceiling((n-2)/2)! - 6floor(n/2)!*ceiling((n-2)/2)!)^(-1)).