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A145883
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Triangle read by rows: T(n,k) is the number of odd permutations of {1,2,...,n} having k descents. (n>=1, k>=1).
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1
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0, 1, 2, 1, 6, 6, 12, 36, 12, 28, 155, 147, 29, 1, 56, 605, 1208, 586, 64, 1, 120, 2160, 7800, 7800, 2160, 120, 240, 7320, 44160, 78000, 44160, 7320, 240, 496, 23947, 227623, 655039, 655315, 227569, 23893, 517, 1, 992, 76305, 1102068, 4868556
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Number of entries in row n is ceil(C(n,2)/2) - ceil(C(n-2,2/2).
Sum of entries in row n is A001710(n) for n>=2.
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REFERENCES
| R. P. Stanley, Binomial posets, Moebius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.
J. Shareshian and M. L. Wachs, q-Eulerian polynomials: excedance number and major index, Electronic Research Announcements of the Amer. Math. Soc., 13 (2007), 33-45.
S. Tanimoto, A study of Eulerian numbers for permutations in the alternating group, Integers, Electronic J. of Combinatorial Number Theory, 6 (2006), #A31.
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FORMULA
| In the Shareshian and Wachs reference (p. 35) a q-analog of the exponential g.f. of the Eulerian polynomials is given for the joint distribution of (inv, des) (see also the Stanley reference). The Maple program given below makes use of this function by considering its odd part.
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EXAMPLE
| T(4,2)=6 because we have 1432, 3142, 3214, 4312, 4231 and 3421.
Triangle starts:
0;
1;
2,1;
6,6;
12,36,12;
28,155,147,29,1;
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MAPLE
| for n to 11 do qbr := proc (m) options operator, arrow; sum(q^i, i = 0 .. m-1) end proc; qfac := proc (m) options operator, arrow; product(qbr(j), j = 1 .. m) end proc; Exp := proc (z) options operator, arrow; sum(q^binomial(m, 2)*z^m/qfac(m), m = 0 .. 19) end proc; g := (1-t)/(Exp(z*(t-1))-t); gser := simplify(series(g, z = 0, 17)); a[n] := simplify(qfac(n)*coeff(gser, z, n)); b[n] := (a[n]-subs(q = -q, a[n]))*1/2; P[n] := sort(subs(q = 1, b[n])) end do; 0; for n to 11 do seq(coeff(P[n], t, j), j = 1 .. ceil((1/2)*binomial(n, 2))-ceil((1/2)*binomial(n-2, 2))) end do; # yields sequence in triangular form
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CROSSREFS
| A001710, A145882
Sequence in context: A117965 A111646 A117753 * A062820 A113336 A113979
Adjacent sequences: A145880 A145881 A145882 * A145884 A145885 A145886
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 11 2008
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