A174704 -id:A174704 - cp's OEIS frontend

A187817 Number of permutations p of {1,...,n} such that exactly two elements of {p(1),...,p(i-1)} are between p(i) and p(i+1) for all i from 3 to n-1.

1, 1, 2, 6, 4, 4, 4, 4, 8, 12, 20, 32, 52, 104, 188, 344, 616, 1116, 2232, 4236, 8084, 15212, 28760, 57520, 111512, 216804, 417560, 806440, 1612880, 3162132, 6209192, 12113136, 23670168, 47340336, 93411704, 184494460, 362693224, 713767712, 1427535424

Offset: 0

Alois P. Heinz, Oct 03 2013

nonn

			a(4) = 4: 2314, 2341, 3214, 3241.
a(5) = 4: 23514, 32514, 34152, 43152.
a(6) = 4: 341625, 346152, 431625, 436152.
a(7) = 4: 3471625, 4371625, 4517263, 5417263.
a(8) = 8: 34716258, 43716258, 45182736, 45817263, 54182736, 54817263, 56283741, 65283741.
a(9) = 12: 348172596, 438172596, 451827369, 459182736, 541827369, 549182736, 561928374, 569283741, 651928374, 659283741, 672938514, 762938514.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A174700, A174701, A174702, A174703, A174704, A174705, A174706, A174707, A174708, A185030, A216837.

b:= proc(u, o) option remember; `if`(u+o<3, (u+o)!,
      `if`(o>2, b(sort([o-3, u+2])[]), 0)+
      `if`(u>2, b(sort([u-3, o+2])[]), 0))
    end:
a:= n-> `if`(n=0, 1, add(b(sort([j-1, n-j])[]), j=1..n)):
seq(a(n), n=0..40);

b[u_, o_] := b[u, o] = If[u + o < 3, (u + o)!,
     If[o > 2, b @@ Sort[{o - 3, u + 2}], 0] +
     If[u > 2, b @@ Sort[{u - 3, o + 2}], 0]];
a[n_] := If[n == 0, 1, Sum[b @@ Sort[{j - 1, n - j}], {j, 1, n}]];
a /@ Range[0, 40] (* Jean-François Alcover, Mar 15 2021, after Alois P. Heinz *)

A249631 Number of permutations p of {1,...,n} such that |p(i+1)-p(i)| < k, k=2,...,n; T(n,k), read by rows.

Original entry on oeis.org

2, 2, 6, 2, 12, 24, 2, 20, 72, 120, 2, 34, 180, 480, 720, 2, 56, 428, 1632, 3600, 5040, 2, 88, 1042, 5124, 15600, 30240, 40320, 2, 136, 2512, 15860, 61872, 159840, 282240, 362880, 2, 208, 5912, 50186, 236388, 773040, 1764000, 2903040, 3628800

Offset: 2

Li-yao Xia, Nov 02 2014

			Triangle starts with n=2:
2;
2,  6;
2, 12,  24;
2, 20,  72, 120;
2, 34, 180, 480, 720;

Li-yao Xia, Triangle of T(n,k) for n=2..10, k=2..n

Cf. A000142, main diagonal, A062119, subdiagonal.
Cf. A003274, A174700, A174701, A174702, 2nd to 5th columns, T(n,k), k=3,4,5,6.
Cf. A174703, A174704, A174705, A174706, A174707, A174708, similar definitions.

a n x = filter (\l -> all (< x) (zipWith (\x y -> abs (x - y)) l (tail l))) (permutations [1 .. n])

isokp(perm, k) = {for (i=1, #perm-1, if (abs(perm[i]-perm[i+1]) >= k, return (0));); return (1);}
tabl(nn) = {for (n=2, nn, for (k=2, n, print1(sum(i=1, n!, isokp(numtoperm(n, i), k)), ", ");); print(););} \\ Michel Marcus, Nov 06 2014

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A187817 Number of permutations p of {1,...,n} such that exactly two elements of {p(1),...,p(i-1)} are between p(i) and p(i+1) for all i from 3 to n-1.

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