A361950 Array read by antidiagonals: T(n,k) = n! * Sum_{s} 2^(Sum_{i=1..k-1} s(i)*s(i+1))/(Product_{i=1..k} s(i)!) where the sum is over all nonnegative compositions s of n into k parts.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 13, 26, 1, 0, 1, 5, 22, 81, 162, 1, 0, 1, 6, 33, 166, 721, 1442, 1, 0, 1, 7, 46, 287, 1726, 9153, 18306, 1, 0, 1, 8, 61, 450, 3309, 24814, 165313, 330626, 1, 0, 1, 9, 78, 661, 5650, 50975, 494902, 4244481, 8488962, 1, 0

Offset

0,8

Comments

T(n,k) corresponds to c(k,n) in the Klarner reference. This is an intermediate step in the computation of the number of labeled weakly graded (ranked) posets. The number of elements in the poset is n and the rank k.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.

Example

Array begins:
======================================================
n/k| 0 1      2       3        4        5        6 ...
---+--------------------------------------------------
0  | 1 1      1       1        1        1        1 ...
1  | 0 1      2       3        4        5        6 ...
2  | 0 1      6      13       22       33       46 ...
3  | 0 1     26      81      166      287      450 ...
4  | 0 1    162     721     1726     3309     5650 ...
5  | 0 1   1442    9153    24814    50975    91866 ...
6  | 0 1  18306  165313   494902  1058493  1957066 ...
7  | 0 1 330626 4244481 13729846 29885567 55363650 ...
  ...
T(3,2) = 26: the nonnegative integer compositions of 3 with 2 parts are (0,3), (1,2), (2,1), (3,0). These contribute, respectively 2^0*3!/(0!*3!) = 1, 2^2*3!/(1!*2!) = 12, 2^2*3!/(2!*1!) = 12, 2^0*3!/(0!*3!) = 1, so T(3,2) = 1 + 12 + 12 + 1 = 26.

Prog

(PARI)
S(M)={matrix(#M, #M, i, j, sum(k=0, i-j, 2^((j-1)*k)*M[i-j+1, k+1])/(j-1)! )}
C(n, m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
R(n)={Mat([Col(serlaplace(p)) | p<-C(n)])}
{ my(A=R(6)); for(i=1, #A, print(A[i, ])) }

Crossrefs

Columns k=0..7 are A000007, A000012, A047863, A000684, A001827, A001828, A001829, A001830.

Rows 0..2 are A000012, A000027, A028872(n+1).

The unlabeled version is A361952.

Cf. A361951.

Sequence in context: A118345 A292804 A118350 * A183135 A294042 A287316

Adjacent sequences: A361947 A361948 A361949 * A361951 A361952 A361953

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 31 2023

Status

approved