cp's OEIS Frontend

A143947 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the right-to-left minima is k (1 <= k <= n*(n+1)/2).

%I A143947 #24 Jan 05 2024 17:59:48
%S A143947 1,0,1,1,0,0,2,1,2,1,0,0,0,6,2,3,7,2,3,1,0,0,0,0,24,6,8,14,27,10,9,14,
%T A143947 3,4,1,0,0,0,0,0,120,24,30,46,68,142,41,53,50,73,23,17,23,4,5,1,0,0,0,
%U A143947 0,0,0,720,120,144,204,270,436,834,260,256,351,310,463,148,145,118,148,40
%N A143947 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the right-to-left minima is k (1 <= k <= n*(n+1)/2).
%C A143947 Row n contains n(n+1)/2 entries, first n-1 of which are 0. Sum of entries in row n = n! = A000142(n).
%C A143947 Sum of entries in column n = A143948(n).
%C A143947 T(n,n) = (n-1)!.
%C A143947 Sum_{k=n..n(n+1)/2} k*T(n,k) = A001705(n).
%H A143947 Alois P. Heinz, <a href="/A143947/b143947.txt">Rows n = 1..50, flattened</a>
%F A143947 Generating polynomial of row n is (n-1+t)(n-2+t^2)(n-3+t^3)...(1+t^(n-1))t^n.
%e A143947 T(4,6) = 3 because we have 4132, 3142 and 2143 with right-to-left minima at positions 2 and 4.
%e A143947 Triangle starts:
%e A143947   1;
%e A143947   0,  1,  1;
%e A143947   0,  0,  2,  1,  2,  1;
%e A143947   0,  0,  0,  6,  2,  3,  7,  2,  3,  1;
%e A143947   0,  0,  0,  0, 24,  6,  8, 14, 27, 10,  9, 14,  3,  4,  1;
%e A143947   ...
%p A143947 P:=proc(n) options operator, arrow: sort(expand(product(t^(n-j)+j,j=0..n-1))) end proc: for n to 7 do seq(coeff(P(n),t,i),i=1..(1/2)*n*(n+1)) end do; # yields sequence in triangular form
%t A143947 T[n_] := CoefficientList[Product[n-k+t^k, {k, 1, n-1}] t^(n-1), t];
%t A143947 Array[T, 10] // Flatten (* _Jean-François Alcover_, Feb 14 2021 *)
%Y A143947 Cf. A000142, A000217, A001705, A143946, A143948.
%Y A143947 T(n,2n) gives A368678.
%Y A143947 Row maxima give A367594.
%K A143947 nonn,tabf
%O A143947 1,7
%A A143947 _Emeric Deutsch_, Sep 22 2008