login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A262125 Number T(n,k) of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 15
1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 5, 3, 1, 0, 0, 16, 24, 4, 1, 0, 0, 61, 101, 57, 5, 1, 0, 0, 272, 862, 311, 123, 6, 1, 0, 0, 1385, 4743, 3857, 778, 254, 7, 1, 0, 0, 7936, 47216, 27589, 14126, 1835, 514, 8, 1, 0, 0, 50521, 322039, 355751, 111811, 47673, 4189, 1031, 9, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Rows n = 0..100, flattened

FORMULA

T(n,k) = A262124(n,k) - A262124(n,k-1) for k>0, T(n,0) = A262124(n,0).

EXAMPLE

T(4,1) = 5: 1324, 1423, 2314, 2413, 3412.

T(4,2) = 3: 1243, 1342, 2341.

T(4,3) = 1: 1234.

Triangle T(n,k) begins:

  1;

  1,    0;

  0,    1,    0;

  0,    2,    1,    0;

  0,    5,    3,    1,   0;

  0,   16,   24,    4,   1,   0;

  0,   61,  101,   57,   5,   1, 0;

  0,  272,  862,  311, 123,   6, 1, 0;

  0, 1385, 4743, 3857, 778, 254, 7, 1, 0;

MAPLE

b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, x^c,

      (p-> add(coeff(p, x, i)*x^max(i, c), i=0..degree(p)))(add(

       b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))

    end:

T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, i), i=0..n)

             )(add(b(j-1, n-j, 0), j=1..n))):

seq(T(n), n=0..10);

MATHEMATICA

b[u_, o_, c_] := b[u, o, c] = If[c<0, 0, If[u+o==0, x^c, Sum[Coefficient[ #, x, i]*x^Max[i, c], {i, 0, Exponent[#, x]}]]& @ Sum[b[u-j, o-1+j, c-1], {j, 1, u}] + Sum[b[u+j-1, o-j, c+1], {j, 1, o}]];

T[n_] := If[n==0, {1}, Table[Coefficient[#, x, i], {i, 0, n}]]& @ Sum[b[j-1, n-j, 0], {j, 1, n}];

T /@ Range[0, 10] // Flatten (* Jean-Fran├žois Alcover, Jan 19 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=1-10 give: A000111 (for n>1), A320976, A320977, A320978, A320979, A320980, A320981, A320982, A320983, A320984.

Row sums give A000246.

T(2n,n) gives A262127.

Cf. A258829, A262124.

Sequence in context: A079508 A057150 A185663 * A105868 A267163 A265163

Adjacent sequences:  A262122 A262123 A262124 * A262126 A262127 A262128

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 11 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 18 22:37 EDT 2022. Contains 353826 sequences. (Running on oeis4.)