A114219 Number triangle T(n,k) = (k-(k-1)*0^(n-k))*[k<=n].

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 2, 3, 4, 1, 0, 1, 2, 3, 4, 5, 1, 0, 1, 2, 3, 4, 5, 6, 1, 0, 1, 2, 3, 4, 5, 6, 7, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1

Offset

0,9

Comments

Row sums are n*(n-1)/2+1 (essentially A000124). Diagonal sums are A114220. First difference triangle of A077028, when this is viewed as a number triangle.

From R. J. Mathar, Mar 22 2013: (Start)

The matrix inverse is

1;

0, 1;

0, -1, 1;

0, 1, -2, 1;

0, -2, 4, -3, 1;

0, 6, -12, 9, -4, 1;

0, -24, 48, -36, 16, -5, 1;

0, 120, -240, 180, -80, 25, -6, 1;

0, -720, 1440, -1080, 480, -150, 36, -7, 1;

... apparently related to A208058. (End)

Number of permutations of length n avoiding simultaneously the patterns 132 and 321 with k left-to-right maxima (resp., right-to-left minima). A left-to-right maximum (resp., right-to-left minimum) in a permutation p(1)p(2)...p(n) is a position i such that p(j) < p(i) for all j < i (resp., p(j) > p(i) for all j > i). - Sergey Kitaev, Nov 18 2023

Links

Table of n, a(n) for n=0..65.

Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.

Formula

G.f.: (1-x-u*x + 2u*x^2)/((1-x)(1-u*x)^2), where x records length and u records left-to-right maxima (or right-to-left minima). - Sergey Kitaev, Nov 18 2023

Example

Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 1, 2, 1;
  0, 1, 2, 3, 1;
  0, 1, 2, 3, 4, 1;
  0, 1, 2, 3, 4, 5, 1;
  0, 1, 2, 3, 4, 5, 6, 1;
  0, 1, 2, 3, 4, 5, 6, 7, 1;
  ...

Maple

A114219 := proc(n, k)
    if k < 0 or k > n then
        0;
    elif n = k then
        1;
    else
        k ;
    end if;
end proc: # R. J. Mathar, Mar 22 2013

Crossrefs

Cf. A000124, A114220, A077028.

Sequence in context: A174203 A353642 A308151 * A119339 A037835 A116433

Adjacent sequences: A114216 A114217 A114218 * A114220 A114221 A114222

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 18 2005

Status

approved