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A303697 Number T(n,k) of permutations p of [n] whose difference between sum of up-jumps and sum of down-jumps equals k; triangle T(n,k), n>=0, min(0,1-n)<=k<=max(0,n-1), read by rows. 6
1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 4, 5, 4, 5, 4, 1, 1, 11, 19, 19, 20, 19, 19, 11, 1, 1, 26, 82, 100, 101, 100, 101, 100, 82, 26, 1, 1, 57, 334, 580, 619, 619, 620, 619, 619, 580, 334, 57, 1, 1, 120, 1255, 3394, 4339, 4420, 4421, 4420, 4421, 4420, 4339, 3394, 1255, 120, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

LINKS

Alois P. Heinz, Rows n = 0..125, flattened

FORMULA

T(n,0) = A153229(n) for n > 0.

T(n,1) = A005165(n-1) for n > 0.

T(n+1,n-1) = A000295(n).

T(n,k) = T(n,-k).

Sum_{k=0..n-1} k^2 * T(n,k) = A001720(n+2) for n>1.

EXAMPLE

Triangle T(n,k) begins:

:                               1                             ;

:                               1                             ;

:                          1,   0,   1                        ;

:                     1,   1,   2,   1,   1                   ;

:                1,   4,   5,   4,   5,   4,   1              ;

:           1,  11,  19,  19,  20,  19,  19,  11,   1         ;

:      1,  26,  82, 100, 101, 100, 101, 100,  82,  26,  1     ;

:  1, 57, 334, 580, 619, 619, 620, 619, 619, 580, 334, 57, 1  ;

MAPLE

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,

      add(b(u-j, o+j-1)*x^(-j), j=1..u)+

      add(b(u+j-1, o-j)*x^( j), j=1..o)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(

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

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

MATHEMATICA

b[u_, o_] := b[u, o] = Expand[If[u+o == 0, 1,

     Sum[b[u-j, o+j-1] x^-j, {j, 1, u}] +

     Sum[b[u+j-1, o-j] x^j, {j, 1, o}]]];

T[0] = {1};

T[n_] := x^n Sum[b[j-1, n-j], {j, 1, n}] // CoefficientList[#, x]& // Rest;

T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Feb 20 2021, after Alois P. Heinz *) *)

CROSSREFS

Row sums give A000142.

Cf. A000295, A001720, A005165, A008292, A081285, A153229, A291680, A291684, A291722, A316292, A316293, A321316.

Sequence in context: A284414 A140274 A095231 * A342413 A202019 A295685

Adjacent sequences:  A303694 A303695 A303696 * A303698 A303699 A303700

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Apr 28 2018

STATUS

approved

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Last modified December 6 06:14 EST 2021. Contains 349563 sequences. (Running on oeis4.)