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A010028
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Triangle read by rows: T(n,k) is one-half the number of permutations of length n with exactly n-k rising or falling successions, for n >= 1, 1 <= k <= n. T(1,1) = 1 by convention.
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9
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1, 1, 0, 1, 2, 0, 1, 5, 5, 1, 1, 8, 24, 20, 7, 1, 11, 60, 128, 115, 45, 1, 14, 113, 444, 835, 790, 323, 1, 17, 183, 1099, 3599, 6423, 6217, 2621, 1, 20, 270, 2224, 11060, 32484, 56410, 55160, 23811, 1, 23, 374, 3950, 27152, 118484, 325322, 554306, 545135, 239653
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OFFSET
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1,5
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COMMENTS
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(1/2) times number of permutations of 12...n such that exactly n-k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
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REFERENCES
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F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
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LINKS
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FORMULA
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For n>1, coefficient of t^(n-k) in S[n](t) defined in A002464, divided by 2.
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EXAMPLE
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Triangle T(n,k) begins:
1;
1, 0;
1, 2, 0;
1, 5, 5, 1;
1, 8, 24, 20, 7;
1, 11, 60, 128, 115, 45;
1, 14, 113, 444, 835, 790, 323;
1, 17, 183, 1099, 3599, 6423, 6217, 2621;
...
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MAPLE
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S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
[n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
-(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
end:
T:= (n, k)-> ceil(coeff(S(n), t, n-k)/2):
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MATHEMATICA
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S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2]-(1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4]]]; T[n_, k_] := Ceiling[Coefficient[S[n], t, n-k]/2]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)
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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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