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A360288 Number T(n,k) of permutations of [n] whose excedance set is the k-th finite subset of positive integers in standard order; triangle T(n,k), n>=0, 0<=k<=ceiling(2^(n-1))-1, read by rows.

%I A360288 #48 Dec 02 2024 09:30:20
%S A360288 1,1,1,1,1,3,1,1,1,7,3,7,1,3,1,1,1,15,7,31,3,17,7,15,1,7,3,7,1,3,1,1,
%T A360288 1,31,15,115,7,69,31,115,3,37,17,69,7,37,15,31,1,15,7,31,3,17,7,15,1,
%U A360288 7,3,7,1,3,1,1,1,63,31,391,15,245,115,675,7,145,69
%N A360288 Number T(n,k) of permutations of [n] whose excedance set is the k-th finite subset of positive integers in standard order; triangle T(n,k), n>=0, 0<=k<=ceiling(2^(n-1))-1, read by rows.
%C A360288 The list of finite subsets of positive integers in standard statistical (or Yates) order begins: {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, ... cf. A048793, A048794.
%C A360288 The excedance set of permutation p of [n] is the set of indices i with p(i)>i, a subset of [n-1].
%C A360288 All terms are odd.
%H A360288 Alois P. Heinz, <a href="/A360288/b360288.txt">Rows n = 0..15, flattened</a>
%H A360288 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%H A360288 Wikipedia, <a href="https://en.wikipedia.org/wiki/Yates_analysis#Yates_Order">Yates Order</a>
%F A360288 Sum_{k=0..2^(n-1)-1} (k+1) * T(n,k) = A029767(n) for n>=1.
%F A360288 Sum_{k=0..2^(n-1)-1} (2^n-1-k) * T(n,k) = A355258(n+1) for n>=1.
%e A360288 T(5,4) = 3: there are 3 permutations of [5] with excedance set {3} (the 4th subset in standard order): 12435, 12534, 12543.
%e A360288 Triangle T(n,k) begins:
%e A360288   1;
%e A360288   1;
%e A360288   1,  1;
%e A360288   1,  3, 1,  1;
%e A360288   1,  7, 3,  7, 1,  3, 1,  1;
%e A360288   1, 15, 7, 31, 3, 17, 7, 15, 1, 7, 3, 7, 1, 3, 1, 1;
%e A360288   ...
%p A360288 b:= proc(s, t) option remember; (m->
%p A360288       `if`(m=0, x^(t/2), add(b(s minus {i}, t+
%p A360288       `if`(i<m, 2^i, 0)), i=s)))(nops(s))
%p A360288     end:
%p A360288 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n}, 0)):
%p A360288 seq(T(n), n=0..7);
%t A360288 b[s_, t_] := b[s, t] = Function [m, If[m == 0, x^(t/2), Sum[b[s ~Complement~ {i}, t + If[i < m, 2^i, 0]], {i, s}]]][Length[s]];
%t A360288 T[n_] := CoefficientList[b[Range[n], 0], x];
%t A360288 Table[T[n], {n, 0, 7}]  // Flatten (* _Jean-François Alcover_, Feb 13 2023, after _Alois P. Heinz_ *)
%Y A360288 Columns k=0-1 give: A000012, A000225(n-1) for n>=1.
%Y A360288 Row sums give A000142.
%Y A360288 Row lengths are A011782.
%Y A360288 See A152884, A360289 for similar triangles.
%Y A360288 Cf. A000027, A029767, A048793, A048794, A263756, A329369.
%K A360288 nonn,look,tabf
%O A360288 0,6
%A A360288 _Alois P. Heinz_, Feb 01 2023