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A108723 Triangle read by rows: T(n,k) is number of permutations of [n] with ascending runs consisting of consecutive integers and having k fixed points. 0
1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 6, 0, 1, 0, 1, 10, 4, 0, 1, 0, 1, 26, 0, 4, 0, 1, 0, 1, 42, 16, 0, 4, 0, 1, 0, 1, 106, 0, 16, 0, 4, 0, 1, 0, 1, 170, 64, 0, 16, 0, 4, 0, 1, 0, 1, 426, 0, 64, 0, 16, 0, 4, 0, 1, 0, 1, 682, 256, 0, 64, 0, 16, 0, 4, 0, 1, 0, 1, 1706, 0, 256, 0, 64, 0, 16, 0, 4, 0, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

T(n,0)=A061547(n). Sum of row n is 2^(n-1) (n>=1).

LINKS

Table of n, a(n) for n=0..90.

FORMULA

T(n, 0)=(3/8)2^n + (1/24)(-2)^n - 2/3; T(n, n)=1; T(n, k)=2^(n-k-2) if k>0 and n-k is even; T(n, k)=0 if k>0 and n-k is odd or if k>n.

EXAMPLE

T(3,0)=2 because we have (23)(1) and (3)(12); T(3,1)=1 because we have (3)(2)(1); T(3,3)=1 because we have (123) (the ascending runs are shown between parentheses).

Triangle begins:

1;

0,1;

1,0,1;

2,1,0,1;

6,0,1,0,1;

10,4,0,1,0,1;

MAPLE

T:=proc(n, k) if k=n then 1 elif k>n then 0 elif k=0 then 3*2^n/8+(-2)^n/24-2/3 elif k>0 and n-k mod 2 = 0 then 2^(n-k-2) else 0 fi end: for n from 0 to 13 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

CROSSREFS

Cf. A061547.

Sequence in context: A180782 A213597 A302978 * A291584 A352451 A349618

Adjacent sequences:  A108720 A108721 A108722 * A108724 A108725 A108726

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 21 2005

STATUS

approved

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Last modified June 29 10:08 EDT 2022. Contains 354912 sequences. (Running on oeis4.)