OFFSET
0,5
COMMENTS
A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67.
Number of entries in row n is 1+floor(n/2).
LINKS
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, Series A, Vol. 99, No. 2 (2002), pp. 345-357.
FORMULA
T(n,k) = Sum(binomial(k+2i,k)*binomial(n/2+i-1,k+2i-1)*[d(k+2i)+d(k+2i-1)], i=0..n/2-k) if n is even,
T(n,k) = Sum(binomial(k+2i+1,k)*binomial(n/2-1/2+i,k+2i)*[d(k+2i+1)+d(k+2i)], i=0..n/2-1/2-k) if n is odd,
Here d(i)=A000166(i) are the derangement numbers.
Sum of entries in row n = n! = A000142(n).
T(2n+1,n) = d(n+2).
Sum(k*T(n,k), k>=0) = A180195(n-1).
EXAMPLE
T(3,1)=2 because we have (23)1 and 3(12) (the blocks of even length are shown between parentheses).
Triangle starts:
1;
1;
1,1;
4,2;
13,10,1;
63,48,9;
356,293,68,3;
MAPLE
d[ -1] := 0: d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if `mod`(n, 2) = 0 then sum(binomial(k+2*i, k)*binomial((1/2)*n+i-1, k+2*i-1)*(d[k+2*i]+d[k+2*i-1]), i = 0 .. (1/2)*n-k) else sum(binomial(k+2*i+1, k)*binomial((1/2)*n+i-1/2, k+2*i)*(d[k+2*i+1]+d[k+2*i]), i = 0 .. (1/2)*n-1/2-k) end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 09 2010
STATUS
approved