A194354 Up-down permutations on [n] whose peaks have k rises.

1, 1, 0, 1, 0, 2, 0, 3, 2, 0, 8, 8, 0, 15, 38, 8, 0, 48, 176, 48, 0, 105, 692, 540, 48, 0, 384, 3584, 3584, 384, 0, 945, 13884, 26204, 9104, 384, 0, 3840, 78848, 188416, 78848, 3840, 0, 109, 315294, 1194380, 997576, 181280, 3840

Offset

0,6

Comments

Triangle read by rows: T(n,k) is the number of up-down permutations (p(i),i=1..n) on [n] such that the subpermutation of peaks (p(2),p(4),p(6),...) consists of k decreasing runs, equivalently, has k ascents where the first entry of a nonempty permutation is conventionally considered to be an ascent.

For n>=1, T(n,k) is nonzero only for 1 <= k <= n/2.

References

L. Carlitz, Enumeration of up-down permutations by number of rises, Pacific Journal of Mathematics, Vol. 45, no.1, 1973, 49-58.

Links

Table of n, a(n) for n=0..48.

Formula

Carlitz's recurrence underlies the Mathematica code below.

Example

Table begins
\ k.0....1.....2.....3.....4
n
0 |.1
1 |.1
2 |.0....1
3 |.0....2
4 |.0....3.....2
5 |.0....8.....8
6 |.0...15....38.....8
7 |.0...48...176....48
8 |.0..105...692...540....48
9 |.0..384..3584..3584...384
The up-down permutation 1 9 3 10 6 8 2 5 4 7 is counted by T(10,3) because the subpermutation of peaks splits into 3 decreasing runs: 9, 10 8 5, 7.
T(4,1)=3 counts 1423, 2413, 3412.

Mathematica

Clear[A];
A[m_, r_]/; 0<=m<=1 := If[r==0, 1, 0];
A[m_, r_]/; m>=2 && (r<1 || r>m/2) := 0;
A[m_, r_]/; m>=2 && 1<=r<=m/2 && EvenQ[m] := A[m, r] = Module[{n=m/2},
  Sum[Binomial[2n-1, 2k+1]A[2k+1, s]A[2n-2k-2, r-s], {k, 0, n-2}, {s, 0, r}] + A[2n-1, r-1] ];
A[m_, r_]/; m>=2 && 1<=r<=m/2 && OddQ[m] := A[m, r] = Module[{n=(m-1)/2},
  Sum[Binomial[2n, 2k+1]A[2k+1, s]A[2n-2k-1, r-s], {k, 0, n-2}, {s, 0, r}] + 2n A[2n-1, r-1] ];
Table[A[m, r], {m, 0, 12}, {r, 0, m/2}]

Crossrefs

Row sums are A000111. Column k=1 is the double factorials A006882.

Sequence in context: A362789 A154752 A271868 * A156776 A292108 A352910

Adjacent sequences: A194351 A194352 A194353 * A194355 A194356 A194357

Keyword

nonn,tabl

Author

David Callan, Aug 23 2011

Status

approved