OFFSET
0,5
COMMENTS
The Lah numbers with level 3 T(n, k) count ordered 3-tuples (pi(1), pi(2), pi(3)) of partitions of the set {1, ..., n} into k linearly ordered blocks (lists, for short) such that bl(pi(1)) = bl(pi(2) = bl(pi(3))) where for i = {1, 2, 3} and pi(i) = b(1)^i, b(2)^i, ..., b(k)^i, where b(1)^i, b(2)^i, ..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min(b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).
The Lah numbers with level 3 T(n, k) are the Lah numbers with higher level for s=3. More generally, the Lah numbers with higher level count ordered s-tuples (pi(1), pi(2), ..., pi(s)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that bl(pi(1)) = bl(pi(2)) = ... = bl(pi(s)) where for i = {1, 2, ..., s} and pi(i) = {b(1)^i, b(2)^i, ..., b(k)^i}, where b(1)^i, b(2)^i,..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min(b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).
LINKS
A. Žigon Tankosič, The Lah Numbers with Higher Level and the Lah Numbers of Order s, arXiv:2510.26842 [math.GM], 2025.
FORMULA
Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^3*T(n-1, k) with boundary conditions T(n,n) = 1, T(n,0) = T(0,n) = 0 for n > 0, T(n, k) = 0 for n < k.
Explicit formula: T(n, k) = Sum_{1 <= j(1) < j(2) < ... < j(n-k) <= n} (2*j(1)-2)^3 * (2*j(2)-3)^3 * ... * (2*j(n-k)-(n-k+1))^3.
EXAMPLE
Triangle begins:
1
0, 1
0, 8, 1
0, 216, 72, 1
0, 13824, 9216, 288, 1
0, 1728000, 2004480, 108000, 800, 1
0, 373248000, 689264640, 57300480, 691200, 1800, 1
0, 128024064000, 353276743680, 42461314560, 748500480, 3087000, 3528, 1
...
MAPLE
T := proc(n, k) option remember; `if`(n = k, 1, `if`(k < 0 or n < k, 0, T(n - 1, k - 1) + (n +k- 1)^3*T(n - 1, k))); end proc;
for n from 0 to 10 do
seq(T(n, k), k = 0 .. n);
end do;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Aleks Zigon Tankosic, Nov 05 2025
STATUS
approved