OFFSET
0,6
COMMENTS
Row lengths are A011782(n).
Every row begins and ends with a 1 because there is exactly 1 n-permutation whose descent set is the empty set and there is exactly 1 n-permutation whose descent set is {1,2,...,n-1}, namely the identity permutation and its reverse.
LINKS
Alois P. Heinz, Rows n = 0..15, flattened
EXAMPLE
T(5,5) = 6 because there are 6 permutations of [5] whose descent set is {1,2}: (3,2,1,4,5), (4,2,1,3,5), (4,3,1,2,5), (5,2,1,3,4), (5,3,1,2,4), (5,4,1,2,3).
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 2, 2, 1;
1, 3, 5, 3, 3, 5, 3, 1;
1, 4, 9, 9, 4, 6, 16, 11, 11, 16, 6, 4, 9, 9, 4, 1;
...
MAPLE
T:= proc(n) option remember; local b, i, l; l:=
map(x-> add(2^(i-1), i=x), [seq(combinat[choose](
[$1..n-1], i)[], i=0..n-1)]); h(0):=0;
for i to nops(l) do h(l[i]):= (i-1) od: b:=
proc(u, o, t) option remember; `if`(u+o=0, x^h(t),
add(b(u-j, o+j-1, t), j=1..u)+
add(b(u+j-1, o-j, t+2^(u+o-1)), j=1..o))
end; (p->
seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2))
end:
seq(T(n), n=0..7); # Alois P. Heinz, Feb 03 2023
MATHEMATICA
f[list_] := (-1)^(Length[list] + 1) Apply[Multinomial, list];
Table[g[S_] :=Abs[Total[Map[f, Map[Differences, Map[Prepend[#, 0] &, Map[Append[#, n] &, Subsets[S]]]]]]]; Map[g, Subsets[Range[n - 1]]], {n, 1, 5}] // Grid
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Jun 26 2020
EXTENSIONS
T(0,0)=1 prepended by Alois P. Heinz, Sep 08 2020
STATUS
approved