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%I A368401 #32 Dec 28 2023 15:37:37
%S A368401 1,1,1,1,1,2,3,1,3,7,11,2,1,4,12,28,53,12,10,1,5,18,52,135,289,84,72,
%T A368401 58,6,1,6,25,84,257,734,1825,524,564,496,422,60,42,1,7,33,125,429,
%U A368401 1407,4545,12983,3520,3976,4292,3950,3422,790,486,330,24
%N A368401 Number T(n,k) of permutations of [n] whose sum of cycle maxima minus cycle minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows.
%H A368401 Alois P. Heinz, <a href="/A368401/b368401.txt">Rows n = 0..21, flattened</a>
%H A368401 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A368401 Sum_{k=0..A002620(n)} k * T(n,k) = A002538(n-1) for n >= 1.
%e A368401 T(3,0) = 1: (1)(2)(3).
%e A368401 T(3,1) = 2: (12)(3), (1)(23).
%e A368401 T(3,2) = 3: (123), (132), (13)(2).
%e A368401 Triangle T(n,k) begins:
%e A368401   1;
%e A368401   1;
%e A368401   1, 1;
%e A368401   1, 2,  3;
%e A368401   1, 3,  7, 11,   2;
%e A368401   1, 4, 12, 28,  53,  12,   10;
%e A368401   1, 5, 18, 52, 135, 289,   84,  72,  58,   6;
%e A368401   1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42;
%e A368401   ...
%p A368401 b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
%p A368401       b(n-1, s)+add(b(n-1, subs(h=h+[0, 1], s)), h=s), 0)+
%p A368401       `if`(n>k+1, b(n-1, {s[], [n,1]}), 0)+add(h[2]!*expand(
%p A368401       x^(h[1]-n)*b(n-1, s minus {h})), h=s))(nops(s)))
%p A368401     end:
%p A368401 T:= (n, k)-> coeff(b(n, {}), x, k):
%p A368401 seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);
%Y A368401 Row sums give A000142.
%Y A368401 Cf. A002538, A002620, A124327, A143946, A367955, A368338.
%K A368401 nonn,look,tabf
%O A368401 0,6
%A A368401 _Alois P. Heinz_, Dec 22 2023

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