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A134433
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which the last entry of the first increasing run is equal to k (1 <= k <= n).
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2
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1, 0, 2, 0, 1, 5, 0, 2, 6, 16, 0, 6, 16, 33, 65, 0, 24, 60, 114, 196, 326, 0, 120, 288, 522, 848, 1305, 1957, 0, 720, 1680, 2952, 4632, 6850, 9786, 13700, 0, 5040, 11520, 19800, 30336, 43710, 60672, 82201, 109601
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OFFSET
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1,3
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COMMENTS
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T(n,n) = A000522(n-1) (number of arrangements of {1,2,...,n-1}).
T(n,2) = (n-2)! for n >= 3.
Sum_{k=1..n} k*T(n,k) = A056542(n+1).
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LINKS
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FORMULA
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T(n,k) = Sum_{j=1..k-1} (n-j-1)!*(k-j)*binomial(k-1,j-1) for k < n;
T(n,n) = (n-1)!*Sum_{j=0..n-1} 1/j!.
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EXAMPLE
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T(4,3)=6 because we have 3124, 3142, 3214, 3241, 1324 and 2314.
Triangle starts:
1;
0, 2;
0, 1, 5;
0, 2, 6, 16;
0, 6, 16, 33, 65;
0, 24, 60, 114, 196, 326;
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MAPLE
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T:=proc(n, k): if k < n then sum(factorial(k-1)*factorial(n-j-1)/(factorial(j-1)*factorial(k-j-1)), j=1..k-1) elif k = n then factorial(n-1)*(sum(1/factorial(j), j = 0 .. n-1)) else 0 end if end proc: for n to 9 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form
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MATHEMATICA
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Table[If[k < n, Sum[(n - j - 1)!*(k - j)*Binomial[k - 1, j - 1], {j, k - 1}], (n - 1)!*Sum[1/j!, {j, 0, n - 1}]], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Nov 15 2019 *)
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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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