A002630 -id:A002630

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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)).

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..56.

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

FORMULA

Riordan gives a recurrence.

EXAMPLE

Triangle begins:

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;

...

CROSSREFS

Columns give A002628, A002629, A002630.

Row sums give A000142.

KEYWORD

nonn,tabf,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited and extended by Max Alekseyev, Sep 05 2010

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

STATUS

approved

A247109

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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)!)))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Tani Akinari, Nov 21 2014

STATUS

approved

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