This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A152884 #89 May 29 2025 13:33:45
%S A152884 1,1,1,1,1,3,1,1,1,7,3,1,7,3,1,1,1,15,7,3,1,31,17,7,7,3,1,15,7,3,1,1,
%T A152884 1,31,15,7,3,1,115,69,37,15,31,17,7,7,3,1,115,69,31,37,17,7,15,7,3,1,
%U A152884 31,15,7,3,1,1,1,63,31,15,7,3,1,391,245,145,77,31,115,69,37,15,31,17,7,7,3,1
%N A152884 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} with excedance set equal to the k-th subset of {1,2,...,n-1} (n>=0, 0<=k<=ceiling(2^(n-1))-1). The subsets of {1,2,...,n-1} are ordered according to size, while the subsets of the same size follow the lexicographic order.
%C A152884 For example, the eight subsets of {1,2,3} are ordered as empty,1,2,3,12,13,23,123. The excedance set of a permutation p of {1,2,...,n} is the set of indices i such that p(i)>i; it is a subset of {1,2,...,n-1}.
%C A152884 Row n contains ceiling(2^(n-1)) entries.
%C A152884 Sum of entries in row n is n! (A000142).
%C A152884 The given Maple program yields the term of the sequence corresponding to a specified permutation size n and a specified excedance set A.
%C A152884 All terms are odd. - _Alois P. Heinz_, Jan 31 2023
%H A152884 Alois P. Heinz, <a href="/A152884/b152884.txt">Rows n = 0..15, flattened</a>
%H A152884 R. Ehrenborg and E. Steingrimsson, <a href="http://dx.doi.org/10.1006/aama.1999.0671">The excedance set of a permutation</a>, Advances in Appl. Math., 24, 284-299, 2000.
%F A152884 T(n,k) = A000225(n-k) = 2^(n-k) - 1 for n>k>0. - _Alexander R. Povolotsky_, May 14 2025
%e A152884 T(5,3) = 3 because the 3rd subset of {1,2,3,4} is {3} and the permutations of {1,2,3,4,5} with excedance set {3} are 12435, 12534 and 12543.
%e A152884 T(5,4) = 1: 12354 (4th subset of {1,2,3,4} is {4}).
%e A152884 Triangle starts:
%e A152884 k=0 1 2 3 4 5 6 7 8 ...
%e A152884 n=0: 1;
%e A152884 n=1: 1;
%e A152884 n=2: 1, 1;
%e A152884 n=3: 1, 3, 1, 1;
%e A152884 n=4: 1, 7, 3, 1, 7, 3, 1, 1;
%e A152884 n=5: 1, 15, 7, 3, 1, 31, 17, 7, 7, 3, 1, 15, 7, 3, 1, 1;
%e A152884 ...
%p A152884 n := 7: A := {1, 2, 4}: with(combinat): P := permute(n): EX := proc (p) local S, i: S := {}: for i to n-1 do if i < p[i] then S := `union`(S, {i}) else end if end do: S end proc: ct := 0: for j to factorial(n) do if EX(P[j]) = A then ct := ct+1 else end if end do: ct;
%p A152884 # second Maple program:
%p A152884 T:= proc(n) option remember; uses combinat; local b, i, l;
%p A152884 l:= map(x-> {x[]}, [seq(choose([$1..n-1], i)[], i=0..n-1)]):
%p A152884 for i to nops(l) do h(l[i]):=i od:
%p A152884 b:= proc(s, l) option remember; (m->
%p A152884 `if`(m=0, x^h(l), add(b(s minus {i}, {l[],
%p A152884 `if`(i<m, i, [][])}), i=s)))(nops(s))
%p A152884 end; (p->
%p A152884 seq(coeff(p, x, i), i=1..degree(p)))(b({$1..n}, {}))
%p A152884 end: T(0):=1:
%p A152884 seq(T(n), n=0..8); # _Alois P. Heinz_, Jan 29 2023
%Y A152884 Row sums are A000142.
%Y A152884 See A360288, A360289 for similar triangles.
%Y A152884 Cf. A000225, A011782, A082185, A136126, A193360, A329369 (another version).
%K A152884 nonn,look,tabf
%O A152884 0,6
%A A152884 _Emeric Deutsch_, Jan 13 2009
%E A152884 T(0,0)=1 prepended and indexing adapted by _Alois P. Heinz_, Jan 29 2023