

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(n1).


160



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
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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^42*x^2x1)/(x^62*x^3+1).
For n > 0, also the number of compositions of n with k + 1 runs.  Gus Wiseman, Apr 10 2020


LINKS



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(ni, 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[ni, 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 (* JeanFrançois 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*(n1)/3]]}] (* Gus Wiseman, Apr 10 2020 *)


CROSSREFS

Columns k=010 give: A000005 (for n>0), 2*A002133, A244714, A244715, A244716, A244717, A244718, A244719, A244720, A244721, A244722.
The version counting adjacent equal parts is A106356.
The version for ascents/descents is A238343.
The version for weak ascents/descents is A333213.
The kth composition in standardorder has A124762(k) adjacent equal parts, A124767(k) maximal runs, A333382(k) adjacent unequal parts, and A333381(k) maximal antiruns.


KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



