OFFSET
0,9
COMMENTS
LINKS
R. Chapman and L. K. Williams, A conjecture of Stanley on alternating permutations, The Electronic J. of Combinatorics, 14, 2007, #N16.
R. P. Stanley, Alternating permutations and symmetric functions, J. Comb. Theory A 114 (3) (2007) 436-460.
FORMULA
The row generating polynomials can be obtained from Proposition 6.1 of the Stanley reference (see the Maple program).
EXAMPLE
T(5,2)=3 because we have 15243, 14352, and 25341.
Triangle starts:
1;
0, 1;
0, 0, 1;
1, 1;
2, 2, 1;
6, 6, 3, 1;
22, 22, 12, 4, 1;
102, 102, 51, 15, 2;
MAPLE
fo := exp(E*(arctan(q*t)-arctan(t)))/(1-E*t): fe := sqrt((1+q^2*t^2)/(1+t^2))*exp(E*(arctan(q*t)-arctan(t)))/(1-E*t): foser := simplify(series(fo, t = 0, 18)): feser := simplify(series(fe, t = 0, 18)): Q := proc (n) if `mod`(n, 2) = 1 then coeff(foser, t, n) else coeff(feser, t, n) end if end proc: for n from 0 to 16 do Q(n) end do: g := sec(x)+tan(x): gser := series(g, x = 0, 20): for n from 0 to 18 do a[n] := factorial(n)*coeff(gser, x, n) end do: for n from 0 to 15 do P[n] := sort(subs({E = a[1], E^2 = a[2], E^3 = a[3], E^4 = a[4], E^5 = a[5], E^6 = a[6], E^7 = a[7], E^8 = a[8], E^9 = a[9], E^10 = a[10], E^11 = a[11], E^12 = a[12], E^13 = a[13], E^14 = a[14], E^15 = a[15], E^16 = a[16]}, Q(n))) end do: 1; 0, 1; 0, 0, 1; 1, 1; 2, 2, 1; for n from 5 to 13 do seq(coeff(P[n], q, j), j = 0 .. 1+floor((1/2)*n)) end do;
MATHEMATICA
nmax = 10;
fo = Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1 - e*t);
fe = Sqrt[(1+q^2 t^2)/(1+t^2)]*Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1-e*t);
Q[n_] := If[OddQ[n], SeriesCoefficient[fo, {t, 0, n}], SeriesCoefficient[fe, {t, 0, n}]] // Expand;
b[n_] := n!*SeriesCoefficient[Sec[x] + Tan[x], {x, 0, n}];
P[n_] := (Q[n] /. e^k_Integer :> b[k]) /. e :> b[1] // Expand;
T[n_, k_] := Coefficient[P[n], q, k];
Table[CoefficientList[P[n], q], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Jul 24 2018, from Maple *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 06 2009
STATUS
approved