cp's OEIS Frontend

A372208 Triangle read by rows, (3, 3)-Lah numbers.

%I A372208 #31 Aug 18 2025 00:10:04
%S A372208 1,216,1,74088,728,1,37933056,604800,1728,1,27653197824,642733056,
%T A372208 2904768,3456,1,27653197824000,883130895360,5662172160,10497600,6200,
%U A372208 1,36806406303744000,1553703385006080,13322923130880,34467586560,31422600,10296,1,63601470092869632000,3450292743162101760,38111804456140800,129651027770880,163174556160,82006848,16128,1
%N A372208 Triangle read by rows, (3, 3)-Lah numbers.
%C A372208 The (3, 3)-Lah numbers 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 the numbers 1, 2, 3 are in distinct lists, and 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).
%C A372208 The (3, 3)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=3 and r=3. More generally, the (m, r)-Lah numbers count ordered m-tuples (pi(1), pi(2), ..., pi(m)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, ..., r are in distinct lists, and bl(pi(1)) = bl(pi(2)) = ... = bl(pi(m)) where for i = {1, 2, ..., m} 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 A372208 A. Žigon Tankosič, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Tankosic/tank2.html">The (l,r)-Lah Numbers</a>, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).
%F A372208 Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^3*T(n-1, k).
%F A372208 Explicit formula: T(n, k) = Sum_{4 <= j(1) < j(2) < ... < j(n-k) <= n} (2j(1)-2)^3 * (2j(2)-3)^3 * ... * (2j(n-k)-(n-k+1))^3.
%F A372208 Special cases:
%F A372208 T(n, k) = 0 for n < k or k < 3, [corrected by _Paolo Xausa_, Jun 11 2024]
%F A372208 T(n, n) = 1,
%F A372208 T(n, 3) = (A143498(n, 3))^3 = ((n+2)!)^3/1728000,
%F A372208 T(n, n-1) = 2^3 * Sum_{j=3..n-1} j^3.
%e A372208 Triangle begins:
%e A372208               1;
%e A372208             216,             1;
%e A372208           74088,           728,           1;
%e A372208        37933056,        604800,        1728,         1;
%e A372208     27653197824,     642733056,     2904768,      3456,     1;
%e A372208  27653197824000,  883130895360,  5662172160,  10497600,  6200,  1.
%e A372208   ...
%e A372208 An example for T(4, 3). The corresponding partitions are
%e A372208 pi(1) = {(1),(2),(3,4)},
%e A372208 pi(2) = {(1),(2),(4,3)},
%e A372208 pi(3) = {(1),(3),(2,4)},
%e A372208 pi(4) = {(1),(3),(4,2)},
%e A372208 pi(5) = {(1,4),(2),(3)},
%e A372208 pi(6) = {(4,1),(2),(3)}, since A143498 for n=4, k=3 equals 6. Sets of their block leaders are bl(pi(1)) = bl(pi(2)) = bl(pi(3)) = bl(pi(4)) = bl(pi(5)) = bl(pi(6)) = {1,2,3}. Compute the number of ordered 3-tuples (i.e., ordered pairs) of partitions pi(1), pi(2), ..., pi(6) such that partitions in the same pair share the same set of block leaders. As there are six partitions with the set of block leaders equal to {1,2,3}, T(4, 3) = 6^3 = 216.
%p A372208 T:= proc(n, k) option remember; `if`(k<3 or k>n, 0,
%p A372208       `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^3*T(n-1, k)))
%p A372208     end:
%p A372208 seq(seq(T(n, k), k=3..n), n=3..10);
%t A372208 A372208[n_, k_] := A372208[n, k] = Which[n < k || k < 3, 0, n == k, 1, True, A372208[n-1, k-1] + (n+k-1)^3*A372208[n-1, k]];
%t A372208 Table[A372208[n, k], {n, 3, 10}, {k, 3, n}] (* _Paolo Xausa_, Jun 11 2024 *)
%o A372208 (Python)
%o A372208 A372208 = lambda n, k: 0 if (k < 3 or k > n) else (1 if (n == 3 and k == 3) else (A372208(n-1, k-1) + ((n + k - 1)**3) * A372208(n-1, k)))
%o A372208 print([A372208(n, k) for n in range(3, 11) for k in range(3, n+1)])
%Y A372208 Cf. A143498, A371081, A371259, A371277.
%K A372208 nonn,tabl
%O A372208 3,2
%A A372208 _Aleks Zigon Tankosic_, Apr 22 2024