1,2

Row n has ceil(n/2) terms.

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009: (Start)

Sum of entries in row n is n! = A000142(n).

T(n,0)=2^(n-1)=A000079(n-1).

T(n,1)=A000431(n).

T(n,2)=A000487(n).

T(n,3)=A000517(n).

T(2n,n-1)=T(2n+1,n)=A000182(n+1) (the tangent numbers).

Sum(k*T(n,k), k>=0)=n!(n-2)/3=A090672(n-1).

(End)

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.3.

W. O. Kermack and A. G. McKendrick, "Some distributions associated with a randomly arranged set of numbers", Proc. Royal Soc. of Edinburgh 67 (1937), 332-376.

W. O. Kermack and A. G. McKendrick, "Some properties of points arranged on a M\"obius surface", Mathematical Gazette 22 (1938), 66-72.

Louis W. Shapiro, Wen-Jin Woan and Seyoum Getu, "Runs, slides and moments", SIAM J. Algebraic and Discrete Methods 4 (1983), 461.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.46. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009]

E.g.f.: G(t,z)=[exp(bz)-exp(az)]/[b*exp(az)-a*exp(bz)], where a+b=2 and ab=t, i.e. a=1+sqrt(1-t), b=1-sqrt(1-t) (see the Goulden-Jackson reference). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009]

1; 2; 4,2; 8,16; 16,88,16; 32,416,272; 64,1824,2880,272; ... T(3,1)=2 because we have 132 and 231.

Comment from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009: The Maple program yields (by straightforward counting) the generating polynomial of the row n specified in the program.

n := 8: with(combinat): P := permute(n): st := proc (p) local ct, j: ct := 0: for j from 2 to nops(p)-1 do if p[j-1] < p[j] and p[j+1] < p[j] then ct := ct+1 else end if end do: ct end proc: sort(add(t^st(P[j]), j = 1 .. factorial(n))); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009]

a := 1+sqrt(1-t): b := 1-sqrt(1-t): G := (exp(b*z)-exp(a*z))/(b*exp(a*z)-a*exp(b*z)): Gser := simplify(series(G, z = 0, 15)): for n to 12 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n to 12 do seq(coeff(P[n], t, j), j = 0 .. ceil((1/2)*n)-1) end do; # yields sequence in triangular form [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009]

(C++) #include <vector> #include <iostream> using namespace std ; int peaks(const vector<int> & perm) { int pks=0 ; for(int i=1 ; i < perm.size()-1; i++) if ( perm[i]>perm[i+1] && perm[i]> perm[i-1] ) pks++ ; return pks ; } int main(int argc, char *argv[]) { int n=1 ; if ( argc > 1 ) n=atoi(argv[1]) ; int nmax = n+12 ; if ( argc > 2 ) nmax=atoi(argv[2]) ; for(; n < nmax ; n++) { const int kmax = (n+1)/2 ; vector<int> Tnk(kmax) ; vector<int> perm(n) ; for(int i=0 ; i < n ; i++) perm[i]=i+1 ; int pks = peaks(perm) ; Tnk[pks] ++ ; while ( next_permutation(perm.begin(), perm.end()) ) { pks = peaks(perm) ; Tnk[pks]++ ; } for (int i=0 ; i < Tnk.size() ; i++) cout << Tnk[i] << ", " ; } } - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 26 2007

Column k=1 yields A000431, column k=2 yields A000487, column k=3 yields A000517.

Cf. A000111, A101280.

A090672, A000182 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009]

Sequence in context: A114655 A051288 A120434 this_sequence A058942 A072866 A061393

Adjacent sequences: A008300 A008301 A008302 this_sequence A008304 A008305 A008306

tabf,nonn

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2004

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 26 2007

Corrected by Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009