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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

%I #3 Mar 30 2012 17:36:05

%S 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,

%T 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,

%U 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

%N 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.

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

%F 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.

%e 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).

%e Triangle begins:

%e 1;

%e 0,1;

%e 1,0,1;

%e 2,1,0,1;

%e 6,0,1,0,1;

%e 10,4,0,1,0,1;

%p 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

%Y Cf. A061547.

%K nonn,tabl

%O 0,7

%A _Emeric Deutsch_, Jun 21 2005