A307269
Number of permutations p of [n] such that |p(i) - p(i-1)| is in {2,5} for all i from 2 to n.
Original entry on oeis.org
1, 1, 0, 0, 0, 0, 2, 14, 12, 8, 28, 58, 44, 120, 254, 226, 344, 932, 1262, 1380, 2958, 5006, 5632, 9496, 18204, 23756, 32758, 59992, 90494, 118740, 196318, 320814, 437270, 653770, 1077580, 1570054, 2233920, 3551168, 5426452, 7714408, 11709864
Offset: 0
a(6) = 2: 246135, 531642.
a(7) = 14: 1357246, 1642753, 2461357, 2753164, 3164275, 3572461, 4275316, 4613572, 5316427, 5724613, 6135724, 6427531, 7246135, 7531642.
a(8) = 12: 13572468, 13864275, 16427538, 16835724, 42753168, 42753861, 57246138, 57246831, 83164275, 83572461, 86135724, 86427531.
a(9) = 8: 168357249, 168357942, 249753168, 249753861, 861357249, 861357942, 942753168, 942753861.
-
b:= proc(s, l) option remember; `if`(s={}, 1, add(
`if`(abs(l-j) in {2, 5}, b(s minus {j}, j), 0), j=s))
end:
a:= proc(n) option remember; if n=0 then 1 else
add(b({$1..n} minus {j}, j), j=1..n) fi
end:
seq(a(n), n=0..20);
-
b[s_, l_] := b[s, l] = If[s == {}, 1, Sum[If[MemberQ[{2, 5}, Abs[l - j]], b[s ~Complement~ {j}, j], 0], {j, s}]];
a[n_] := a[n] = If[n==0, 1, Sum[b[Range[n] ~Complement~ {j}, j], {j, n}]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 23 2021, after Alois P. Heinz *)
A328648
Number of permutations p of [n] such that |p(i) - p(i-1)| is in {2,7} for all i from 2 to n.
Original entry on oeis.org
1, 1, 0, 0, 0, 0, 0, 0, 2, 18, 12, 0, 12, 62, 76, 32, 44, 162, 600, 714, 386, 550, 2514, 5320, 4140, 3336, 8626, 24722, 33428, 27110, 34812, 96210, 200322, 220360, 213368, 376178, 894780, 1400578, 1473944, 1810538, 3653304, 7170370, 9467970
Offset: 0
a(8) = 2: 24681357, 75318642.
a(9) = 18: 135792468, 186429753, 246813579, 297531864, 318642975, 357924681, 429753186, 468135792, 531864297, 579246813, 642975318, 681357924, 753186429, 792468135, 813579246, 864297531, 924681357, 975318642.
a(10) = 12: 135792468(10), 13(10)8642975, 186429753(10), 18(10)3579246, 579246813(10), 5792468(10)31, 642975318(10), 6429753(10)81, (10)318642975, (10)357924681, (10)813579246, (10)864297531.
-
b:= proc(s, l) option remember; `if`(s={}, 1, add(`if`(l=0
or abs(l-j) in {2, 7}, b(s minus {j}, j), 0), j=s))
end:
a:= n-> b({$1..n}, 0):
seq(a(n), n=0..20);
-
b[s_, l_] := b[s, l] = If[s == {}, 1, Sum[If[l == 0 || MemberQ[{2, 7}, Abs[l - j]], b[s ~Complement~ {j}, j], 0], {j, s}]];
a[n_] := b[Range[n], 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 23 2021, after Alois P. Heinz *)
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.
Original entry on oeis.org
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
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.
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
Triangle starts with n=2:
2;
2, 6;
2, 12, 24;
2, 20, 72, 120;
2, 34, 180, 480, 720;
Cf.
A003274,
A174700,
A174701,
A174702, 2nd to 5th columns, T(n,k), k=3,4,5,6.
-
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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