A129534 -id:A129534 - cp's OEIS frontend

A129535 Number of permutations of 1,...,n with at least one pair of adjacent consecutive entries (i.e., of the form k(k+1) or (k+1)k), n >= 2.

2, 6, 22, 106, 630, 4394, 35078, 315258, 3149494, 34620010, 415222566, 5395737242, 75516784982, 1132471183626, 18115911832390, 307919970965434, 5541804787940598, 105282261866132138, 2105441434230129254, 44210612765653749210, 972564180363044943766

Offset: 2

Emeric Deutsch, May 05 2007

nonn

Column 1 of A129534. a(n) = n! - A002464(n).

Column k=2 of A322481.

			a(4)=22 because 3142 and 2413 are the only permutations of 1,2,3,4 with no adjacent consecutive entries.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.40.

Alois P. Heinz, Table of n, a(n) for n = 2..450

Cf. A002464, A129534, A322481.

E:=x->sum(n!*x^n,n=0..35): G:=E(x)-E(x*(1-x)/(1+x)): Gser:=series(G,x=0,30): seq(coeff(Gser,x,n),n=2..23);

G.f.: E(x) - E(x(1-x)/(1+x)), where E(x) = Sum_{n>=0} n!*x^n.
a(n) = n! - Sum_{k=1..n} ((-1)^(n-k)*k!*Sum_{i=0..n-k} binomial(i+k-1, k-1)*binomial(k, n-i-k)), n > 0. - Vladimir Kruchinin, Sep 08 2010
D-finite with recurrence a(n) +2*(-n+1)*a(n-1) +(n^2-2*n-2)*a(n-2) +(-n^2+7*n-14)*a(n-3) -(n-3)*(n-5)*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 26 2022

A064482 Triangle read by rows: T(n,k) (n >= 2, 1<=k<=n-1) is the number of permutations p of 1,...,n with max(|p(i)-p(i-1)|, i=2..n) = k.

Original entry on oeis.org

2, 2, 4, 2, 10, 12, 2, 18, 52, 48, 2, 32, 146, 300, 240, 2, 54, 372, 1204, 1968, 1440, 2, 86, 954, 4082, 10476, 14640, 10080, 2, 134, 2376, 13348, 46012, 97968, 122400, 80640, 2, 206, 5704, 44274, 186202, 536652, 990960, 1139040, 725760, 2, 312, 13278, 145216, 742940, 2655004, 6562128, 10847520, 11692800, 7257600

Offset: 2

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 05 2001

T(n,n-1) = A052849; sum(T(n,k),k=1..n-1) = A000142.

			Triangle T(n,k) begins:
  2;
  2,   4;
  2,  10,   12;
  2,  18,   52,    48;
  2,  32,  146,   300,   240;
  2,  54,  372,  1204,  1968,  1440;
  2,  86,  954,  4082, 10476, 14640,  10080;
  2, 134, 2376, 13348, 46012, 97968, 122400, 80640;

Alois P. Heinz, Rows n = 2..18, flattened

Cf. A052849, A000142, A129534.

More terms from Naohiro Nomoto, Dec 04 2001

More terms from R. J. Mathar, Oct 11 2007

A322255 Triangle T(n,k) giving the number of permutations of 1..n with no adjacent elements within k in value, for n >= 2, 1 <= k <= floor(n/2).

Original entry on oeis.org

2, 6, 24, 2, 120, 14, 720, 90, 2, 5040, 646, 32, 40320, 5242, 368, 2, 362880, 47622, 3984, 72, 3628800, 479306, 44304, 1496, 2, 39916800, 5296790, 521606, 25384, 160, 479001600, 63779034, 6564318, 399848, 6056, 2, 6227020800, 831283558, 88422296, 6231544, 161136, 352

Offset: 2

Seiichi Manyama, Dec 01 2018

			Irregular triangle starts:
n\k|       1       2      3     4  5
---+---------------------------------
2  |       2;
3  |       6;
4  |      24,      2;
5  |     120,     14;
6  |     720,     90,     2;
7  |    5040,    646,    32;
8  |   40320,   5242,   368,    2;
9  |  362880,  47622,  3984,   72;
10 | 3628800, 479306, 44304, 1496, 2;

Cf. A000142, A002464, A127697, A129534, A179957-A179967.

T(n,k) = Sum_{j=k..floor(n/2)} A129534(n,j). - Alois P. Heinz, May 20 2023

cp's OEIS Frontend

A129535 Number of permutations of 1,...,n with at least one pair of adjacent consecutive entries (i.e., of the form k(k+1) or (k+1)k), n >= 2.

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A064482 Triangle read by rows: T(n,k) (n >= 2, 1<=k<=n-1) is the number of permutations p of 1,...,n with max(|p(i)-p(i-1)|, i=2..n) = k.

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A322255 Triangle T(n,k) giving the number of permutations of 1..n with no adjacent elements within k in value, for n >= 2, 1 <= k <= floor(n/2).

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