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.

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

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);

Mathematica

b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, x^(Length[s ~Union~  {t}] - 1), Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}], {j, 1, u}] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
T[n_] := With[{p = b[n, 0, 0, {}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)

Crossrefs

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

Row sums give A000142.

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

Sequence in context: A263399 A143507 A172040 * A120557 A092594 A092741

Adjacent sequences: A317324 A317325 A317326 * A317328 A317329 A317330

Keyword

nonn,tabf

Author

Alois P. Heinz, Jul 25 2018

Status

approved