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A317327 Number T(n,k) of permutations of [n] with exactly k distinct lengths of increasing runs; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows. 5
1, 0, 1, 0, 2, 0, 2, 4, 0, 7, 17, 0, 2, 118, 0, 82, 436, 202, 0, 2, 3294, 1744, 0, 1456, 18164, 20700, 0, 1515, 140659, 220706, 0, 50774, 1096994, 2317340, 163692, 0, 2, 10116767, 27136103, 2663928, 0, 3052874, 94670868, 328323746, 52954112, 0, 2, 1021089326, 4317753402, 888178070 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Rows n = 0..60, flattened

FORMULA

T(n*(n+1)/2,n) = A317273(n).

Sum_{k=0..floor((sqrt(1+8*n)-1)/2)} k * T(n,k) = A317328(n).

EXAMPLE

T(4,1) = 7: 1234, 1324, 1423, 2314, 2413, 3412, 4321.

Triangle T(n,k) begins:

  1;

  0,       1;

  0,       2;

  0,       2,        4;

  0,       7,       17;

  0,       2,      118;

  0,      82,      436,       202;

  0,       2,     3294,      1744;

  0,    1456,    18164,     20700;

  0,    1515,   140659,    220706;

  0,   50774,  1096994,   2317340,   163692;

  0,       2, 10116767,  27136103,  2663928;

  0, 3052874, 94670868, 328323746, 52954112;

MAPLE

b:= proc(u, o, t, s) option remember;

      `if`(u+o=0, x^(nops(s union {t})-1),

       add(b(u-j, o+j-1, 1, s union {t}), j=1..u)+

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

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2, {})):

seq(T(n), n=0..16);

CROSSREFS

Columns k=0-1 give: A000007, A317329.

Row sums give A000142.

Cf. A000217, A003056, A097591, A097592, A123125, A218868, A317273.

Sequence in context: A143507 A071961 A172040 * A120557 A092594 A092741

Adjacent sequences:  A317324 A317325 A317326 * A317328 A317329 A317330

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jul 25 2018

STATUS

approved

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Last modified January 27 12:01 EST 2020. Contains 331295 sequences. (Running on oeis4.)