A250260 -id:A250260 - 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 *)

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 *)

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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