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A180196
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Triangle read by rows: T(n,k) is the number of permutations of [n] that have k isolated entries (0 <= k <= n).
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3
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1, 0, 1, 1, 0, 1, 1, 2, 0, 3, 2, 2, 9, 0, 11, 3, 11, 9, 44, 0, 53, 7, 20, 75, 44, 265, 0, 309, 14, 73, 141, 574, 265, 1854, 0, 2119, 35, 170, 737, 1104, 4900, 1854, 14833, 0, 16687, 81, 576, 1863, 7814, 9535, 46353, 14833, 133496, 0, 148329, 216, 1556, 8154, 20704
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OFFSET
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0,8
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COMMENTS
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An entry j of a permutation p is isolated if it is not preceded by j-1 and not followed by j+1. For example, the permutation 23178564 has 2 isolated entries: 1 and 4.
Sum of entries in row n is n! = A000142(n).
T(n,n) = d(n) + d(n-1) = A000255(n-1), where d(i)=A000166(i) are the derangement numbers.
T(n,n-2) = d(n) (n >= 2).
T(n,n-3) = d(n-1) (n >= 3).
Sum_{k=0..n} k*T(n,k) = (n-2)!*(n^3 - 3n^2 + 5n - 4) = A001565(n-2) (n >= 2).
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LINKS
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FORMULA
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T(n,k) = Sum_{j=k+1..floor((n+k)/2)} binomial(n-1-j, j-k-1)*binomial(j,k)*(d(j) + d(j-1)), if k < n;
T(n,n) = d(n) + d(n-1); d(i)=A000166(i) are the derangement numbers.
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EXAMPLE
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T(4,2)=9 because we have 124'3', 1'4'23, 1'342', 3'124', 4'3'12, 2'1'34, 231'4', 4'231', and 342'1' (the isolated entries are marked).
Triangle starts:
1;
0, 1;
1, 0, 1;
1, 2, 0, 3;
2, 2, 9, 0, 11;
3, 11, 9, 44, 0, 53;
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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 k < n then sum(binomial(n-1-j, j-k-1)*binomial(j, k)*(d[j]+d[j-1]), j = k+1 .. floor((1/2)*n+(1/2)*k)) elif k = n then d[n]+d[n-1] else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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