A002630 -id:A002630 - cp's OEIS frontend

A047921 Triangle of numbers a(n,k) = number of permutations on n letters containing k 3-sequences (n >= 0, 0<=k<=max(0,n-2)).

1, 1, 2, 5, 1, 21, 2, 1, 106, 11, 2, 1, 643, 62, 12, 2, 1, 4547, 406, 71, 13, 2, 1, 36696, 3046, 481, 80, 14, 2, 1, 332769, 25737, 3708, 559, 89, 15, 2, 1, 3349507, 242094, 32028, 4414, 640, 98, 16, 2, 1, 37054436, 2510733, 306723, 38893, 5164, 724, 107, 17, 2, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
       1;
       1;
       2;
       5,     1;
      21,     2,    1;
     106,    11,    2,   1;
     643,    62,   12,   2,  1;
    4547,   406,   71,  13,  2,  1;
   36696,  3046,  481,  80, 14,  2, 1;
  332769, 25737, 3708, 559, 89, 15, 2, 1;
  ...

J. Riordan, Permutations without 3-sequences, Bull. Amer. Math. Soc., 51 (1945), 745-748.

Columns give A002628, A002629, A002630.

Row sums give A000142.

Riordan gives a recurrence.

Edited and extended by Max Alekseyev, Sep 05 2010

a(0,0) = a(1,0) = 1 prepended by Alois P. Heinz, Apr 20 2021

A247109 Number of permutations of length n with two 4-sequences.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 11, 65, 433, 3271, 27741, 261231, 2708064, 30671367, 377034018, 5001404982, 71229862678, 1084282429946, 17571257417630, 302064161086250, 5490937395703435, 105243824522368960, 2121386876912041845, 44863116021267642255, 993272322666679219071, 22977273619066571708457

Offset: 1

Tani Akinari, Nov 21 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Cf. A002630 (permutations with two 3-sequences).

Table[Sum[(-1)^k * k*(k-1)/2 * Sum[Sum[Binomial[k-1,p] * Binomial[k-p-1,i-2*p] * Binomial[n-2*k+p-1,n-3*k+i-1] * (n-3*k+i)!,{p,0,k-1}],{i,Max[0,3*k-n],2*(k-1)}],{k,2,n-3}],{n,1,20}] (* Vaclav Kotesovec, Nov 23 2014 after Tani Akinari *)

a(n)=sum(k=2,n-3,(-1)^k*k*(k-1)/2*sum(i=max(0,3*k-n),2*(k-1),sum(p=0,k-1,binomial(k-1,p)*binomial(k-p-1,i-2*p)*binomial(n-2*k+p-1,n-3*k+i-1)*(n-3*k+i)!)))

a(n) ~ n! / n^3. - Vaclav Kotesovec, Nov 23 2014

cp's OEIS Frontend

A047921 Triangle of numbers a(n,k) = number of permutations on n letters containing k 3-sequences (n >= 0, 0<=k<=max(0,n-2)).

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