A390433 - OEIS

%I #16 Nov 11 2025 22:17:08

%S 1,0,1,0,4,1,0,36,20,1,0,576,536,56,1,0,14400,19872,3280,120,1,0,

%T 518400,988128,229792,13000,220,1,0,25401600,63758592,19601280,

%U 1529792,39620,364,1,0,1625702400,5189847552,2023886592,204706112,7235072,101136,560,1

%N Triangle read by rows, Lah numbers with level 2.

%C The Lah numbers with level 2 T(n, k) count ordered 2-tuples (pi(1), pi(2)) of partitions of the set {1, ..., n} into k linearly ordered blocks (lists, for short) such that bl(pi(1)) = bl(pi(2)) where for i = {1, 2} 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).

%C The Lah numbers with level 2 T(n, k) are the Lah numbers with higher level for s=2. 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).

%H A. Žigon Tankosič, <a href="https://arxiv.org/abs/2510.26842">The Lah Numbers with Higher Level and the Lah Numbers of Order s</a>, arXiv:2510.26842 [math.GM], 2025.

%F Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^2*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.

%F Explicit formula: T(n, k) = Sum_{1 <= j(1) < j(2) < ... < j(n-k) <= n} (2*j(1)-2)^2 * (2*j(2)-3)^2 * ... * (2*j(n-k)-(n-k+1))^2.

%e Triangle begins:

%e   1

%e   0, 1

%e   0, 4, 1

%e   0, 36, 20, 1

%e   0, 576, 536, 56, 1

%e   0, 14400, 19872, 3280, 120, 1

%e   0, 518400, 988128, 229792, 13000, 220, 1

%e   0, 25401600, 63758592, 19601280, 1529792, 39620, 364, 1

%e   0, 1625702400, 5189847552, 2023886592, 204706112, 7235072, 101136, 560, 1

%e   0, 131681894400, 520610457600, 250080125184, 31501566720, 1427433280, 27057728, 227136, 816, 1.

%e   ...

%p 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)^2*T(n - 1, k))); end proc;

%p for n from 0 to 10 do

%p     seq(T(n, k), k = 0 .. n); end do;

%Y Cf. A371081, A268434, A390434.

%K nonn,tabl

%O 0,5

%A _Aleks Zigon Tankosic_, Nov 05 2025