A180190 - OEIS

A180190

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).

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, 234, 86, 32, 1, 23633, 9828, 4122, 1744, 702, 226, 64, 1, 214551, 88028, 36320, 14990, 6094, 2154, 614, 128, 1, 2160343, 876852, 357332, 145242, 58468, 21842, 6750, 1714, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Terms obtained by counting with a time-consuming Maple program.

Sum of entries in row n = n! = A000142(n).

T(n,1) = A180191(n).

LINKS

Alois P. Heinz, Rows n = 1..18, flattened

FORMULA

Sum_{k=0..n-1} k * T(n,k) = A018927(n). - Alois P. Heinz, Feb 21 2019

EXAMPLE

T(4,2) = 6 because we have 1324, 4132, 2413, 4213, 2431, and 3241.

Triangle starts:

1, 1;

1, 3, 2;

1, 13, 6, 4;

1, 67, 30, 14, 8;

...

MAPLE

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

# second Maple program:

b:= proc(s, l, m) option remember; `if`(s={}, x^`if`(m=infinity, 0, m),

add(b(s minus {j}, j, `if`(j<l, m, min(m, j-l))), j=s))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b({$1..n}, infinity$2)):

seq(T(n), n=1..10); # Alois P. Heinz, Feb 21 2019

MATHEMATICA

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[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n - 1}]][b[ Range[n], Infinity, Infinity]];

T /@ Range[10] // Flatten (* Jean-François Alcover, Dec 08 2019, after Alois P. Heinz *)

CROSSREFS

Cf. A000142, A018927, A180191.