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A263753 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n with sum of descent tops equal to k.

%I A263753 #34 Apr 23 2025 13:09:34
%S A263753 1,1,1,0,1,1,0,1,3,0,1,1,0,1,3,7,1,3,7,0,1,1,0,1,3,7,16,3,14,17,32,3,
%T A263753 7,15,0,1,1,0,1,3,7,16,34,14,32,69,72,129,32,68,70,118,7,15,31,0,1,1,
%U A263753 0,1,3,7,16,34,77,32,100,149,274,292,496,220,388,536
%N A263753 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n with sum of descent tops equal to k.
%C A263753 Row sums give A000142.
%C A263753 Row lengths are given by A000217 for n>=1. - _Omar E. Pol_, Oct 25 2015
%H A263753 Alois P. Heinz, <a href="/A263753/b263753.txt">Rows n = 0..23, flattened</a>
%H A263753 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000111">The sum of the descent tops of a permutation</a>
%e A263753 Triangle begins:
%e A263753   1;
%e A263753   1;
%e A263753   1,0,1;
%e A263753   1,0,1,3,0,1;
%e A263753   1,0,1,3,7,1,3,7,0,1;
%e A263753   1,0,1,3,7,16,3,14,17,32,3,7,15,0,1;
%e A263753   1,0,1,3,7,16,34,14,32,69,72,129,32,68,70,118,7,15,31,0,1;
%e A263753   ...
%p A263753 b:= proc(s) option remember; (n-> `if`(n=0, 1, expand(
%p A263753       add(b(s minus {j})*`if`(j<n, x^n, 1), j=s))))(nops(s))
%p A263753     end:
%p A263753 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
%p A263753 seq(T(n), n=0..9);  # _Alois P. Heinz_, Oct 25 2015, revised, Jan 31 2023
%t A263753 b[s_] := b[s] = With [{n = Length[s]},If[n == 0, 1, Expand[ Sum[b[s ~Complement~ {j}]*If[j < n, x^n, 1], {j, s}]]]];
%t A263753 T[n_] := With[{p = b[Range[n]]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A263753 Table[T[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Apr 23 2025, after _Alois P. Heinz_ *)
%Y A263753 Cf. A000142, A263756.
%K A263753 nonn,tabf
%O A263753 0,9
%A A263753 _Christian Stump_, Oct 19 2015
%E A263753 One term prepended and one term corrected by _Alois P. Heinz_, Oct 25 2015