OFFSET
0,8
COMMENTS
Let a=a(1)a(2)...a(2n) be a permutation of [2n] written in one line notation i.e. written as a word. T(n,k) is the number of ways to factor the permutations of [2n] into k factors [a(1)...a(j_1)] [a(j_1+1)... a(j_2)]... [a(j_{k-1}+1) ... a(j_k=2n)] so that each factor has even size and descent set {2,4,6,...,j_i - 2} for i ={1,2,...,k}. In other words, the factors are the up-down permutations enumerated in A000364. The factors may be empty.
FORMULA
T(n,k) = Z(P_n,-k) where Z(P_n,k) is the zeta polynomial of an n-interval in the binomial poset of even sized subsets of the positive integers.
EXAMPLE
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 5, 16, 33, 56, 85, ...
0, 61, 272, 723, 1504, 2705, ...
0, 1385, 7936, 25953, 64256, 134185, ...
0, 50521, 353792, 1376643, 3963904, 9451805, ...
T(2,2) = 16. There is 1 factorization into 2 good factors of the permutation 1234=[12][34]. Each of the permutations 1324,1423,2314,2413,3412 can be factored in 3 good ways. For example 1324=[][1324]=[13][24]=[1324][]. So T(2,2) = 1 + 3(5) = 16.
MATHEMATICA
nn = 5; B[n_] := (2 n)!/2^n; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
Table[Table[B[n], {n, 0, nn}] PadRight[CoefficientList[Series[e[-x]^-k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Transpose // Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 07 2021
STATUS
approved