A341268 Square array read by descending antidiagonals. T(n,k) is the number of ways to factor a permutation of [2n] into exactly k good factors, n>=0, k>=0.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 16, 61, 0, 1, 4, 33, 272, 1385, 0, 1, 5, 56, 723, 7936, 50521, 0, 1, 6, 85, 1504, 25953, 353792, 2702765, 0, 1, 7, 120, 2705, 64256, 1376643, 22368256, 199360981, 0

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.

Links

Table of n, a(n) for n=0..44.

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

Cf. A000364 (column k=1), A000182 (column k=2).

Sequence in context: A336686 A371501 A118349 * A011183 A005671 A343605

Adjacent sequences: A341265 A341266 A341267 * A341269 A341270 A341271

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 07 2021

Status

approved