cp's OEIS Frontend

A143946 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the left-to-right maxima is k (1 <= k <= n(n+1)/2).

%I A143946 #35 Dec 19 2023 18:51:54
%S A143946 1,1,0,1,2,0,2,1,0,1,6,0,6,3,2,3,2,1,0,1,24,0,24,12,8,18,8,10,3,6,3,2,
%T A143946 1,0,1,120,0,120,60,40,90,64,50,39,42,23,28,13,10,8,6,3,2,1,0,1,720,0,
%U A143946 720,360,240,540,384,420,234,372,198,208,168,124,98,75,60,35,34,13,16,8,6,3
%N A143946 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the sum of the positions of the left-to-right maxima is k (1 <= k <= n(n+1)/2).
%C A143946 Row n contains n*(n+1)/2 = A000217(n) entries.
%C A143946 Sum of entries in row n = n! = A000142(n).
%H A143946 Alois P. Heinz, <a href="/A143946/b143946.txt">Rows n = 1..50, flattened</a>
%H A143946 I. Kortchemski, <a href="http://arxiv.org/abs/0804.0446">Asymptotic behavior of permutation records</a>, arXiv: 0804.0446 [math.CO], 2008-2009.
%F A143946 T(n,1) = T(n,3) = (n-1)! for n>=2.
%F A143946 Sum_{k=1..n*(n+1)/2} k * T(n,k) = n! * n = A001563(n).
%F A143946 Generating polynomial of row n is t(t^2+1)(t^3+2)...(t^n+n-1).
%F A143946 Sum_{k=1..n*(n+1)/2} (n*(n+1)/2-k) * T(n,k) = A001804(n). - _Alois P. Heinz_, Dec 19 2023
%e A143946 T(4,6)=3 because we have 1243, 1342 and 2341 with left-to-right maxima at positions 1,2,3.
%e A143946 Triangle starts:
%e A143946    1;
%e A143946    1,  0,  1;
%e A143946    2,  0,  2,  1,  0,  1;
%e A143946    6,  0,  6,  3,  2,  3,  2,  1,  0,  1;
%e A143946   24,  0, 24, 12,  8, 18,  8, 10,  3,  6,  3,  2,  1,  0,  1;
%e A143946   ...
%p A143946 P:=proc(n) options operator, arrow: sort(expand(product(t^j+j-1,j=1..n))) 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
%p A143946 # second Maple program:
%p A143946 b:= proc(n) option remember; `if`(n=0, 1,
%p A143946       expand(b(n-1)*(x^n+n-1)))
%p A143946     end:
%p A143946 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p A143946 seq(T(n), n=1..7);  # _Alois P. Heinz_, Aug 05 2020
%t A143946 row[n_] := CoefficientList[Product[t^k + k - 1, {k, 1, n}], t] // Rest;
%t A143946 Array[row, 7] // Flatten (* _Jean-François Alcover_, Nov 28 2017 *)
%Y A143946 Cf. A000142, A000217, A001563, A001804, A143947.
%Y A143946 T(n,n) gives A368246.
%K A143946 nonn,tabf
%O A143946 1,5
%A A143946 _Emeric Deutsch_, Sep 21 2008