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%I #17 Dec 08 2019 07:40:29
%S 1,1,1,1,3,2,1,13,6,4,1,67,30,14,8,1,411,178,80,34,16,1,2921,1236,530,
%T 234,86,32,1,23633,9828,4122,1744,702,226,64,1,214551,88028,36320,
%U 14990,6094,2154,614,128,1,2160343,876852,357332,145242,58468,21842,6750,1714,256
%N Triangle read by rows: T(n,k) is the number of permutations p of [n] for which k is the smallest among the positive differences p(i+1) - p(i); k=0 for the reversal of the identity permutation (0<=k<=n-1).
%C Terms obtained by counting with a time-consuming Maple program.
%C Sum of entries in row n = n! = A000142(n).
%C T(n,1) = A180191(n).
%H Alois P. Heinz, <a href="/A180190/b180190.txt">Rows n = 1..18, flattened</a>
%F Sum_{k=0..n-1} k * T(n,k) = A018927(n). - _Alois P. Heinz_, Feb 21 2019
%e T(4,2) = 6 because we have 1324, 4132, 2413, 4213, 2431, and 3241.
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 3, 2;
%e 1, 13, 6, 4;
%e 1, 67, 30, 14, 8;
%e ...
%p with(combinat): minasc := proc (p) local j, b: for j to nops(p)-1 do if 0 < p[j+1]-p[j] then b[j] := p[j+1]-p[j] else b[j] := infinity end if end do: if min(seq(b[j], j = 1 .. nops(p)-1)) = infinity then 0 else min(seq(b[j], j = 1 .. nops(p)-1)) end if end proc; for n to 10 do P := permute(n): f[n] := sort(add(t^minasc(P[j]), j = 1 .. factorial(n))) end do: for n to 10 do seq(coeff(f[n], t, i), i = 0 .. n-1) end do; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(s, l, m) option remember; `if`(s={}, x^`if`(m=infinity, 0, m),
%p add(b(s minus {j}, j, `if`(j<l, m, min(m, j-l))), j=s))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b({$1..n}, infinity$2)):
%p seq(T(n), n=1..10); # _Alois P. Heinz_, Feb 21 2019
%t b[s_List, l_, m_] := b[s, l, m] = If[s == {}, x^If[m == Infinity, 0, m], Sum[b[s ~Complement~ {j}, j, If[j < l, m, Min[m, j - l]]], {j, s}]];
%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n - 1}]][b[ Range[n], Infinity, Infinity]];
%t T /@ Range[10] // Flatten (* _Jean-François Alcover_, Dec 08 2019, after _Alois P. Heinz_ *)
%Y Cf. A000142, A018927, A180191.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Sep 07 2010
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