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A136123
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k maximal strings of increasing consecutive integers (0<=k<=floor(n/2)).
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0
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1, 1, 1, 1, 3, 3, 11, 12, 1, 53, 56, 11, 309, 321, 87, 3, 2119, 2175, 693, 53, 16687, 17008, 5934, 680, 11, 148329, 150504, 55674, 8064, 309, 1468457, 1485465, 572650, 96370, 5805, 53, 16019531, 16170035, 6429470, 1200070, 95575, 2119
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row n has 1+floor(n/2) terms. Row sums are the factorials (A000142). Column 0 yields A000255. Column 1 yields A001277. Column 2 yields A001278. Column 3 yields A001279. Column 4 yields A001280. Sum(k*T(n,k),k>=0)=(n-2)!*(n^2 - 3n + 3)=A001564(n-2).
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REFERENCES
| F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.
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FORMULA
| G.f.=G(x,t)=Sum(n!*((1-t)*x^2 - x)/((1-t)*x^2-1))^n, n=0..infinity). - Vladeta Jovovic (vladeta(AT)eunet.rs).
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EXAMPLE
| T(3,0)=3 because we have 132, 213 and 321; T(6,3)=3 because we have 125634, 341256, 563412.
Triangle starts:
1;
1;
1,1;
3,3;
11,12,1;
53,56,11;
309,321,87,3;
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MAPLE
| G:=Sum(factorial(n)*(((1-t)*x^2-x)/((1-t)*x^2-1))^n, n=0..infinity): Gser:= simplify(series(G, x=0, 13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, x, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
| Cf. A000142, A000255, A001277, A001278, A001279, A001280, A001564.
Sequence in context: A146583 A146458 A122573 * A045495 A045494 A027416
Adjacent sequences: A136120 A136121 A136122 * A136124 A136125 A136126
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu) and Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 17 2007
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