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A180194
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Triangle read by rows: T(n,k) is the number of permutations of [n] having k blocks of even length.
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2
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1, 1, 1, 1, 4, 2, 13, 10, 1, 63, 48, 9, 356, 293, 68, 3, 2403, 2036, 557, 44, 18655, 16213, 4902, 539, 11, 163988, 145068, 47203, 6356, 265, 1608667, 1442858, 495710, 76833, 4679, 53, 17415725, 15792362, 5659377, 971992, 75490, 1854, 206202408
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OFFSET
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0,5
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COMMENTS
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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).
Sum of entries in row n = n! = A000142(n).
T(2n,n)=d(n)+d(n-1)=A000255(n-1), where d(i)=A000166(n) are the derangement numbers.
T(2n+1,n)=d(n+2).
Sum(k*T(n,k), k>=0) = A180195(n-1).
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REFERENCES
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A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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