login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A145876 Triangle read by rows: T(n,k) is the number of permutations of [n] having k-1 alternating descents (1<=k<=n). The index i is an alternating descent of a permutation p if either i is odd and p(i)>p(i+1), or i is even and p(i)<p(i+1). 5
1, 1, 1, 2, 2, 2, 5, 7, 7, 5, 16, 26, 36, 26, 16, 61, 117, 182, 182, 117, 61, 272, 594, 1056, 1196, 1056, 594, 272, 1385, 3407, 6669, 8699, 8699, 6669, 3407, 1385, 7936, 21682, 46348, 67054, 76840, 67054, 46348, 21682, 7936, 50521, 151853, 350240, 556952 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums are the factorials (A000142).

T(n,1) = T(n,n) = A000111(n) (Euler or up-down numbers).

Sum(k*T(n,k), k=1..n) = (n+1)!/2 = A001710(n+1).

From Peter Bala, Jun 11 2011: (Start)

Koutras has defined generalized Eulerian numbers associated with a sequence of polynomials - the ordinary Eulerian numbers A008292 being associated with the sequence of monomials x^n. The present array is the triangle of Eulerian numbers associated with the sequence of zigzag polynomials Z(n,x) defined in A147309.

See A109449, A147315 and A185424 for the respective analogs of the Pascal triangle, the Stirling numbers of the second kind and the Bernoulli numbers, associated with the sequence of zigzag polynomials.

(End)

REFERENCES

S.-M. Ma and Y.-M. Yeh, Enumeration of permutations by number of alternating descents, Discr. Math., 339 (2015), 1362-1367.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..141, flattened

D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.

M. V. Koutras, Eulerian numbers associated with sequences of polynomials, The Fibonacci Quarterly, 32 (1994), 44-57.

FORMULA

E.g.f.: F(t,u) = t*(1-tan(u*(t-1))-sec(u*(t-1)))/(tan(u*(t-1))+sec(u*(t-1))-t).

From Peter Bala, Jun 11 2011: (Start)

T(n,k) = sum {j = 0..k} (-1)^(k-j)*binomial(n+1,k-j)*Z(n,j), where Z(n,x) are the zigzag polynomials defined in A147309.

Let M denote the triangular array A109449. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of the present array.

(End)

From Vladimir Shevelev, Jul 01 2011:  (Start)

a(2^(2*n-1)-2^(n-1)+1) == 1 (mod 2^n).

If n is odd prime, then a(2*n^2-n+1) == 1 (mod 2*n) and a((n^2-n+2)/2) == (-1)^((n-1)/2).

(End)

EXAMPLE

T(4,3) = 7 because we have 1243, 4123, 1342, 3124, 2134, 2341 and 4321. For example, for p=1342 the alternating descent is {2,3}; indeed, 2 is even and p(2)=3 < p(3)=4, while 3 is odd and p(3)=4 > p(4)=2.

Triangle starts:

1;

1,    1;

2,    2,   2;

5,    7,   7,   5;

16,  26,  36,  26,  16;

61, 117, 182, 182, 117, 61;

MAPLE

F:=t*(1-tan(u*(t-1))-sec(u*(t-1)))/(tan(u*(t-1))+sec(u*(t-1))-t): Fser:= simplify(series(F, u=0, 12)): for n from 0 to 10 do P[n]:=sort(expand(factorial(n)*coeff(Fser, u, n))) end do: for n to 10 do seq(coeff(P[n], t, j), j=1..n) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(

      add(b(u+j-1, o-j, not t)*`if`(t, 1, x), j=1..o)+

      add(b(u-j, o+j-1, not t)*`if`(t, x, 1), j=1..u)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(0, n, true)):

seq(T(n), n=1..12);  # Alois P. Heinz, Nov 18 2013

MATHEMATICA

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u+j-1, o-j, !t]*If[t, 1, x], {j, 1, o}] + Sum[b[u-j, o+j-1, !t]*If[t, x, 1], {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[0, n, True]]; Table[ T[n], {n, 1, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 19 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000142, A000111, A001710, A008292, A109449, A147315, A185424.

Sequence in context: A260338 A023569 A263373 * A240540 A039878 A162145

Adjacent sequences:  A145873 A145874 A145875 * A145877 A145878 A145879

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 22 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 21 00:32 EST 2018. Contains 299388 sequences. (Running on oeis4.)