A306461
Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n]; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 4, 3, 2, 6, 10, 13, 15, 13, 10, 6, 24, 42, 56, 67, 76, 67, 56, 42, 24, 120, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120, 720, 1320, 1824, 2250, 2612, 2921, 3186, 2921, 2612, 2250, 1824, 1320, 720, 5040, 9360, 13080, 16296, 19086, 21514, 23633, 25487, 23633, 21514, 19086, 16296, 13080, 9360, 5040
Offset: 1
The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement sets {p(i)-i, i=1..3}: {0}, {-1,0,1}, {-1,0,1}, {-2,1}, {-1,2}, {-2,0,2}, respectively. Numbers -2 and 2 occur twice, -1 and 1 occur thrice, and 0 occurs four times. So row n=3 is [2, 3, 4, 3, 2].
Triangle T(n,k) begins:
: 1 ;
: 1, 1, 1 ;
: 2, 3, 4, 3, 2 ;
: 6, 10, 13, 15, 13, 10, 6 ;
: 24, 42, 56, 67, 76, 67, 56, 42, 24 ;
: 120, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120 ;
-
b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, {})):
seq(T(n), n=1..8);
# second Maple program:
T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n):
seq(seq(T(n, k), k=1-n..n-1), n=1..9);
-
T[n_, k_] := -Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];
Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 9}] // Flatten (* Jean-François Alcover, Feb 20 2021, after Alois P. Heinz *)
A306512
Number A(n,k) of permutations p of [n] having no index i with |p(i)-i| = k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 9, 1, 1, 2, 3, 5, 44, 1, 1, 2, 6, 9, 21, 265, 1, 1, 2, 6, 14, 34, 117, 1854, 1, 1, 2, 6, 24, 53, 176, 792, 14833, 1, 1, 2, 6, 24, 78, 265, 1106, 6205, 133496, 1, 1, 2, 6, 24, 120, 362, 1554, 8241, 55005, 1334961
Offset: 0
A(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
A(4,1) = 5: 1234, 1432, 3214, 3412, 4231.
A(4,2) = 9: 1234, 1243, 1324, 2134, 2143, 2341, 4123, 4231, 4321.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 2, 2, 2, 2, 2, ...
2, 2, 3, 6, 6, 6, 6, 6, ...
9, 5, 9, 14, 24, 24, 24, 24, ...
44, 21, 34, 53, 78, 120, 120, 120, ...
265, 117, 176, 265, 362, 504, 720, 720, ...
1854, 792, 1106, 1554, 2119, 2790, 3720, 5040, ...
-
A:= proc(n, k) option remember; `if`(k>=n, n!, LinearAlgebra[
Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)=k, 0, 1))))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(s, k) option remember; (n-> `if`(n=0, 1, add(
`if`(abs(i-n)=k, 0, b(s minus {i}, k)), i=s)))(nops(s))
end:
A:= (n, k)-> `if`(k>=n, n!, b({$1..n}, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
A[n_, k_] := If[k > n, n!, Permanent[Table[If[Abs[i-j] == k, 0, 1], {i, 1, n}, {j, 1, n}]]]; A[0, 0] = 1;
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 05 2021, from first Maple program *)
b[s_, k_] := b[s, k] = With[{n = Length[s]}, If[n == 0, 1, Sum[
If[Abs[i-n] == k, 0, b[s ~Complement~ {i}, k]], {i, s}]]];
A[n_, k_] := If[k >= n, n!, b[Range@n, k]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Sep 01 2021, from second Maple program *)
A306524
Number of permutations p of [n] having at least one index i with |p(i)-i| = 2.
Original entry on oeis.org
0, 0, 0, 3, 15, 86, 544, 3934, 32079, 292509, 2952702, 32712087, 394749367, 5155010088, 72440184064, 1090017765544, 17486996858151, 297965879586295, 5374189975316350, 102290550351854445, 2049025241258716927, 43089888746430771294, 949172134240270646352
Offset: 0
a(3) = 3: 231, 312, 321.
a(4) = 15: 1342, 1423, 1432, 2314, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4312.
-
T[n_, k_] := n! - Permanent[Table[If[Abs[i-j] == k, 0, 1], {i, 1, n}, {j, 1, n}]];
a[n_] := If[n == 0, 0, T[n, 2]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 31 2021, after Alois P. Heinz in A306506 *)
A306675
Number of permutations p of [2n] having at least one index i with |p(i)-i| = n.
Original entry on oeis.org
0, 1, 15, 455, 25487, 2293839, 302786759, 55107190151, 13225725636255, 4047072044694047, 1537887376983737879, 710503968166486900119, 392198190427900768865711, 254928823778135499762712175, 192726190776270437820610404327, 167671785975355280903931051764519
Offset: 0
-
b:= proc(n, k) b(n, k):= `if`(k=0, n!, b(n+1, k-1) -b(n, k-1)) end:
a:= n-> (2*n)! -b(0, 2*n):
seq(a(n), n=0..16);
-
b[n_, k_] := b[n, k] = If[k == 0, n!, b[n + 1, k - 1] - b[n, k - 1]];
a[n_] := (2n)! - b[0, 2n];
a /@ Range[0, 16] (* Jean-François Alcover, Apr 02 2021, after Alois P. Heinz *)
A324366
Number of permutations p of [n] having at least one index i with |p(i)-i| = 3.
Original entry on oeis.org
0, 0, 0, 0, 10, 67, 455, 3486, 29296, 272064, 2782768, 31112974, 377989835, 4961822943, 70010940186, 1056948399594, 17002714972374, 290376470114307, 5247488148645251, 100046090532522600, 2006982109206921472, 42259647960510173184, 931935465775791556864
Offset: 0
a(4) = 10: 2341, 2431, 3241, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
A306511
Number of permutations p of [n] having at least one index i with |p(i)-i| = 1.
Original entry on oeis.org
0, 0, 1, 4, 19, 99, 603, 4248, 34115, 307875, 3085203, 33993870, 408482695, 5316309607, 74499953255, 1118421967520, 17907571955927, 304619809031127, 5486197279305911, 104289196264058030, 2086706157642260387, 43838287730208552691, 964790364323910060691
Offset: 0
-
a:= proc(n) option remember; `if`(n<5, [0$2, 1, 4, 19][n+1],
(2*(n^3-8*n^2+20*n-14)*a(n-1)-(n-4)*(n-1)*(n^2-5*n+7)*
a(n-2)-(n-2)*(n^2-7*n+13)*a(n-3)+(n^4-12*n^3+53*n^2
-102*n+71)*a(n-4)+(n-4)*(n^2-5*n+7)*a(n-5))/(n^2-7*n+13))
end:
seq(a(n), n=0..23);
-
a[n_] := n! - Sum[Sum[(-1)^k (i-k)! Binomial[2i-k, k], {k, 0, i}],
{i, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, May 03 2021, after Vaclav Kotesovec in A078480 *)
Showing 1-6 of 6 results.