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A258829 Number T(n,k) of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 8
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 11, 3, 1, 0, 16, 38, 28, 4, 1, 0, 61, 263, 130, 62, 5, 1, 0, 272, 1260, 1263, 340, 129, 6, 1, 0, 1385, 10871, 8090, 4734, 819, 261, 7, 1, 0, 7936, 66576, 88101, 33855, 16066, 1890, 522, 8, 1, 0, 50521, 694599, 724189, 495371, 127538, 52022, 4260, 1040, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

T(n,k) = A262163(n,k) - A262163(n,k-1) for k>0, T(n,0) = A262163(n,0).

EXAMPLE

p = 1432 is counted by T(4,2) because the up-down signature of 0,p = 01432 is 1,1,-1,-1 with partial sums 1,2,1,0.

q = 4321 is not counted by any T(4,k) because the up-down signature of 0,q = 04321 is 1,-1,-1,-1 with partial sums 1,0,-1,-2.

T(4,1) = 5: 2143, 3142, 3241, 4132, 4231.

T(4,2) = 11: 1324, 1423, 1432, 2134, 2314, 2413, 2431, 3124, 3412, 3421, 4123.

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

T(4,4) = 1: 1234.

Triangle T(n,k) begins:

1;

0,    1;

0,    1,     1;

0,    2,     2,    1;

0,    5,    11,    3,    1;

0,   16,    38,   28,    4,   1;

0,   61,   263,  130,   62,   5,   1;

0,  272,  1260, 1263,  340, 129,   6, 1;

0, 1385, 10871, 8090, 4734, 819, 261, 7, 1;

MAPLE

b:= proc(u, o, c, k) option remember;

      `if`(c<0 or c>k, 0, `if`(u+o=0, 1,

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

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

    end:

A:= (n, k)-> b(n, 0$2, k):

T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):

seq(seq(T(n, k), k=0..n), n=0..12);

CROSSREFS

Columns k=0-2 give: A000007, A000111 for n>0, A259213.

Row sums give A258830.

T(2n,n) gives A266947.

Cf. A262124, A262125, A262163.

Sequence in context: A065066 A266291 A064045 * A110314 A152882 A130167

Adjacent sequences:  A258826 A258827 A258828 * A258830 A258831 A258832

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 11 2015

STATUS

approved

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Last modified February 21 04:22 EST 2018. Contains 299389 sequences. (Running on oeis4.)