A249402 -id:A249402 - cp's OEIS frontend

A250261 Number A(n,k) of permutations p of [n] such that p(i) > p(i+1) iff i = 1 + k*m for some m >= 0; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 5, 1, 5, 1, 1, 1, 2, 3, 16, 1, 6, 1, 1, 1, 2, 3, 11, 61, 1, 7, 1, 1, 1, 2, 3, 4, 40, 272, 1, 8, 1, 1, 1, 2, 3, 4, 19, 99, 1385, 1, 9, 1, 1, 1, 2, 3, 4, 5, 78, 589, 7936, 1, 10, 1, 1, 1, 2, 3, 4, 5, 29, 217, 3194, 50521, 1, 11

Offset: 0

Alois P. Heinz, Nov 15 2014

A(n,0) = A(n,k) for k>=n-1 and n>0.

			Square array A(n,k) begins:
  1, 1,    1,   1,   1,   1,  1, 1, 1, ...
  1, 1,    1,   1,   1,   1,  1, 1, 1, ...
  1, 1,    1,   1,   1,   1,  1, 1, 1, ...
  2, 1,    2,   2,   2,   2,  2, 2, 2, ...
  3, 1,    5,   3,   3,   3,  3, 3, 3, ...
  4, 1,   16,  11,   4,   4,  4, 4, 4, ...
  5, 1,   61,  40,  19,   5,  5, 5, 5, ...
  6, 1,  272,  99,  78,  29,  6, 6, 6, ...
  7, 1, 1385, 589, 217, 133, 41, 7, 7, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
J. M. Luck, On the frequencies of patterns of rises and falls, arXiv:1309.7764, 2013
A. Mendes and J. Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page
R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, 2009

Columns k=1-10 give: A000012, A000111, A249402, A250259, A250260, A250262, A250263, A250264, A250265, A250266.

A(n+3,n+1) = A028387(n).

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,
     `if`(t=1, add(b(u-j, o+j-1, irem(t+1, k), k), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, k), k), j=1..o)))
    end:
A:= (n, k)-> b(0, n, 0, `if`(k=0, n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, If[t == 1, Sum[ b[u-j, o+j-1, Mod[t+1, k], k], {j, 1, u}], Sum[ b[u+j-1, o-j, Mod[t+1, k], k], {j, 1, o}] ] ] ; A[n_, k_] := b[0, n, 0, If[k == 0, n, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)

A178963 E.g.f.: (3+2sqrt(3)exp(x/2)sin(sqrt(3)x/2))/(exp(-x)+2exp(x/2)cos(sqrt(3)*x/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 19, 99, 477, 1513, 11259, 74601, 315523, 3052323, 25740261, 136085041, 1620265923, 16591655817, 105261234643, 1488257158851, 17929265150637, 132705221399353, 2172534146099019, 30098784753112329, 254604707462013571, 4736552519729393091, 74180579084559895221, 705927677520644167681, 14708695606607601165843, 256937013876000351610089, 2716778010767155313771539

Offset: 0

N. J. A. Sloane, Dec 31 2010

nonn

According to Mendes and Remmel, p. 56, this is the e.g.f. for 3-alternating permutations.

Alois P. Heinz, Table of n, a(n) for n = 0..500
J. M. Luck, On the frequencies of patterns of rises and falls, arXiv preprint arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.
Peter Luschny, An old operation on sequences: the Seidel transform.
Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), A181936 (m=5).
Cf. A181937, A002115.
Cf. A249402, A249583 (alternative definitions of 3-alternating permutations).

A178963_list := proc(dim) local E,DIM,n,k;
DIM := dim-1; E := array(0..DIM, 0..DIM); E[0,0] := 1;
for n from 1 to DIM do
if n mod 3 = 0 then E[n,0] := 0 ;
   for k from n-1 by -1 to 0 do E[k,n-k] := E[k+1,n-k-1] + E[k,n-k-1] od;
else E[0,n] := 0;
   for k from 1 by 1 to n do E[k,n-k] := E[k-1,n-k+1] + E[k-1,n-k] od;
fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..DIM)] end:
A178963_list(30);  # Peter Luschny, Apr 02 2012
# Alternatively, using a bivariate exponential generating function:
A178963 := proc(n) local g, p, q;
g := (x,z) -> 3*exp(x*z)/(exp(z)+2*exp(-z/2)*cos(z*sqrt(3)/2));
p := (n,x) -> n!*coeff(series(g(x,z),z,n+2),z,n);
q := (n,m) -> if modp(n,m) = 0 then 0 else 1 fi:
(-1)^floor(n/3)*p(n,q(n,3)) end:
seq(A178963(i),i=0..30); # Peter Luschny, Jun 06 2012
# third Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=0, add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 29 2014

max = 30; f[x_] := (E^x*(2*Sqrt[3]*E^(x/2)*Sin[(Sqrt[3]*x)/2] + 3))/(2*E^((3*x)/2)*Cos[(Sqrt[3]*x)/2] + 1); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! // Simplify (* Jean-François Alcover, Sep 16 2013 *)

# uses[A from A181936]
A178963 = lambda n: (-1)^int(is_odd(n//3))*A(3,n)
print([A178963(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017

a(3*n) = A002115(n). - Peter Luschny, Aug 02 2012

A249583 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 2 (mod 3).

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 40, 169, 477, 3194, 19241, 74601, 666160, 5216485, 25740261, 287316122, 2769073949, 16591655817, 222237912664, 2543467934449, 17929265150637, 280180369563194, 3712914075133121, 30098784753112329, 537546603651987424, 8094884285992309261

Offset: 0

Alois P. Heinz, Nov 01 2014

nonn

This is the (UDU)* version of 3-alternating permutations of [n], (U=Up, D=Down).

			a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 9: 13245, 14235, 15234, 23145, 24135, 25134, 34125, 35124, 45123.
a(6) = 40: 132465, 132564, 142365, 142563, 143562, 152364, 152463, 153462, 162354, 162453, 163452, 231465, 231564, 241365, 241563, 243561, 251364, 251463, 253461, 261354, 261453, 263451, 341265, 341562, 342561, 351264, 351462, 352461, 361254, 361452, 362451, 451263, 451362, 452361, 461253, 461352, 462351, 561243, 561342, 562341.
a(7) = 169: 1324657, 1324756, 1325647, ..., 6723514, 6724513, 6734512.

Alois P. Heinz, Table of n, a(n) for n = 0..500
J. M. Luck, On the frequencies of patterns of rises and falls, arXiv:1309.7764, 2013
Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page
R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, 2009

Cf. A178963 (i=0), A249402 (i=1).

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=2, add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..35);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 2, Sum[b[u - j, o + j - 1, Mod[t+1, 3]], {j, 1, u}], Sum[b[u + j - 1, o - j, Mod[t+1, 3]], {j, 1, o}]]];
a[n_] := b[0, n, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

A250259 The number of 4-alternating permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 19, 78, 217, 496, 3961, 25442, 105963, 349504, 3908059, 34227438, 190065457, 819786496, 11785687921, 130746521282, 907546301523, 4835447317504, 84965187064099, 1141012634368398, 9504085749177097, 60283564499562496, 1251854782837499881

Offset: 0

R. J. Mathar, Nov 15 2014

nonn

A sequence a(1),a(2),... is called k-alternating if a(i) > a(i+1) iff i=1 (mod k).

Alois P. Heinz, Table of n, a(n) for n = 0..500
R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240 [math.CO], 2009, page 17.

Cf. A249402 (3-alternating), A065619 (2-alternating), A250260 (5-alternating).

Column k=4 of A250261.

onestep := proc(n::integer,ups::integer,downs::integer,uplen::integer)
    local thisstep,left,doup,tak,ret ;
    option remember;
    left := ups+downs ;
    if left = 0 then
        return 1;
    end if;
    thisstep := n-left+1 ;
    if modp(thisstep-2,uplen+1) = 0 then
        doup := false;
    else
        doup := true;
    end if;
    if doup then
        ret := 0 ;
        for tak from 1 to ups do
            ret := ret+procname(n,ups-tak,downs+tak-1,uplen) ;
        end do:
        return ret ;
    else
        ret := 0 ;
        for tak from 1 to downs do
            ret := ret+procname(n,ups+tak-1,downs-tak,uplen) ;
        end do:
        return ret ;
    end if;
end proc:
downupP := proc(n::integer,uplen::integer)
    local ret,tak;
    if n = 0 then
        return 1;
    end if;
    ret := 0 ;
    for tak from 1 to n do
        ret := ret+onestep(n,n-tak,tak-1,uplen) ;
    end do:
    return ret ;
end proc:
A250259 :=proc(n)
    downupP(n,3) ;
end proc:
seq(A250259(n),n=0..20) ; # R. J. Mathar, Nov 15 2014
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=1, add(b(u-j, o+j-1, irem(t+1, 4)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 4)), j=1..o)))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 15 2014

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 1, Sum[b[u - j, o + j - 1, Mod[t + 1, 4]], {j, 1, u}], Sum[b[u + j - 1, o - j, Mod[t + 1, 4]], {j, 1, o}]]]; a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)

A250260 The number of 5-alternating permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 29, 133, 412, 1041, 2300, 22991, 170832, 822198, 3114489, 10006375, 141705439, 1457872978, 9522474417, 48094772656, 202808749375, 3716808948931, 48860589990687, 403131250565618, 2545098156762649, 13287626090593750

Offset: 0

R. J. Mathar, Nov 15 2014

nonn

A sequence a(1), a(2),... is called k-alternating if a(i) > a(i+1) iff i=1 (mod k).

Alois P. Heinz, Table of n, a(n) for n = 0..500
R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, page 17.

Cf. A065619 (2-alternating), A249402 (3-alternating), A250259 (4-alternating).

Column k=5 of A250261.

# dowupP defined in A250259.
A250260 :=proc(n)
    downupP(n,4) ;
end proc:
seq(A250260(n),n=0..20) ;
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=1, add(b(u-j, o+j-1, irem(t+1, 5)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 5)), j=1..o)))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 15 2014

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, If[t == 1, Sum[b[u-j, o+j-1, Mod[t+1, 5]], {j, 1, u}], Sum[b[u+j-1, o-j, Mod[t+1, 5]], {j, 1, o}]]]; a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 24 2015, after Alois P. Heinz *)

cp's OEIS Frontend

A250261 Number A(n,k) of permutations p of [n] such that p(i) > p(i+1) iff i = 1 + k*m for some m >= 0; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A178963 E.g.f.: (3+2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2))/(exp(-x)+2*exp(x/2)*cos(sqrt(3)*x/2)).

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A249583 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 2 (mod 3).

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A250259 The number of 4-alternating permutations of [n].

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A250260 The number of 5-alternating permutations of [n].

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A178963 E.g.f.: (3+2sqrt(3)exp(x/2)sin(sqrt(3)x/2))/(exp(-x)+2exp(x/2)cos(sqrt(3)*x/2)).