A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1).

1, 1, 2, 2, 2, 3, 4, 1, 2, 10, 4, 4, 12, 14, 2, 2, 22, 29, 10, 1, 4, 26, 56, 36, 6, 3, 34, 100, 86, 31, 2, 4, 44, 148, 200, 99, 16, 1, 2, 54, 230, 374, 278, 78, 8, 6, 58, 322, 680, 654, 274, 52, 2, 2, 74, 446, 1122, 1390, 814, 225, 22, 1, 4, 88, 573, 1796, 2714, 2058, 813, 136, 10, 4, 88, 778, 2694, 4927

Offset

0,3

Comments

Same as A238130, with zeros omitted.

Last elements in rows are 1, 1, 2, 2, 1, 4, 2, 1, 6, 2, 1, 8, ... with g.f. -(x^6+x^4-2*x^2-x-1)/(x^6-2*x^3+1).

For n > 0, also the number of compositions of n with k + 1 runs. - Gus Wiseman, Apr 10 2020

Links

Joerg Arndt and Alois P. Heinz, Rows n = 0..180, flattened

Example

Triangle starts:
00:  1;
01:  1;
02:  2;
03:  2,   2;
04:  3,   4,   1;
05:  2,  10,   4;
06:  4,  12,  14,    2;
07:  2,  22,  29,   10,    1;
08:  4,  26,  56,   36,    6;
09:  3,  34, 100,   86,   31,    2;
10:  4,  44, 148,  200,   99,   16,    1;
11:  2,  54, 230,  374,  278,   78,    8;
12:  6,  58, 322,  680,  654,  274,   52,    2;
13:  2,  74, 446, 1122, 1390,  814,  225,   22,   1;
14:  4,  88, 573, 1796, 2714, 2058,  813,  136,  10;
15:  4,  88, 778, 2694, 4927, 4752, 2444,  618,  77,  2;
16:  5, 110, 953, 3954, 8531, 9930, 6563, 2278, 415, 28, 1;
...
Row n=5 is 2, 10, 4 because in the 16 compositions of 5
##:  [composition]  no. of changes
01:  [ 1 1 1 1 1 ]   0
02:  [ 1 1 1 2 ]   1
03:  [ 1 1 2 1 ]   2
04:  [ 1 1 3 ]   1
05:  [ 1 2 1 1 ]   2
06:  [ 1 2 2 ]   1
07:  [ 1 3 1 ]   2
08:  [ 1 4 ]   1
09:  [ 2 1 1 1 ]   1
10:  [ 2 1 2 ]   2
11:  [ 2 2 1 ]   1
12:  [ 2 3 ]   1
13:  [ 3 1 1 ]   1
14:  [ 3 2 ]   1
15:  [ 4 1 ]   1
16:  [ 5 ]   0
there are 2 with no changes, 10 with one change, and 4 with two changes.

Maple

b:= proc(n, v) option remember; `if`(n=0, 1, expand(
      add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14);

Mathematica

b[n_, v_] := b[n, v] = If[n == 0, 1, Expand[Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)
Table[If[n==0, 1, Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[Split[#]]==k+1&]]], {n, 0, 12}, {k, 0, If[n==0, 0, Floor[2*(n-1)/3]]}] (* Gus Wiseman, Apr 10 2020 *)

Crossrefs

Columns k=0-10 give: A000005 (for n>0), 2*A002133, A244714, A244715, A244716, A244717, A244718, A244719, A244720, A244721, A244722.

Row lengths are A004523.

Row sums are A011782.

The version counting adjacent equal parts is A106356.

The version for ascents/descents is A238343.

The version for weak ascents/descents is A333213.

The k-th composition in standard-order has A124762(k) adjacent equal parts, A124767(k) maximal runs, A333382(k) adjacent unequal parts, and A333381(k) maximal anti-runs.

Cf. A064113, A333214, A333216.

Sequence in context: A338629 A057646 A238892 * A282933 A328576 A052275

Adjacent sequences: A238276 A238277 A238278 * A238280 A238281 A238282

Keyword

nonn,tabf

Author

Joerg Arndt and Alois P. Heinz, Feb 22 2014

Status

approved