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A371259 Triangle read by rows, (2, 3)-Lah numbers.

%I A371259 #46 Aug 17 2025 11:18:18
%S A371259 1,36,1,1764,100,1,112896,9864,200,1,9144576,1099296,34064,344,1,
%T A371259 914457600,142159392,6004512,92200,540,1,110649369600,21385410048,
%U A371259 1156921920,24075712,213700,796,1,15933509222400,3724783667712,248142106368,6573957120,78782912,443744,1120,1
%N A371259 Triangle read by rows, (2, 3)-Lah numbers.
%C A371259 The (2, 3)-Lah numbers 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 the numbers 1, 2, 3 are in distinct lists, and 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 A371259 The (2, 3)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=2 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 A371259 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 A371259 Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^2*T(n-1, k).
%F A371259 Explicit formula: T(n, k) = Sum_{4 <= j(1) < j(2) < ... < j(n-k) <= n} (2j(1)-2)^2 * (2j(2)-3)^2 * ... * (2j(n-k)-(n-k+1))^2.
%F A371259 Special cases:
%F A371259 T(n, k) = 0 for n < k or k < 3, [corrected by _Paolo Xausa_, Jun 11 2024]
%F A371259 T(n, n) = 1,
%F A371259 T(n, 3) = (A143498(n, 3))^2 = ((n+2)!)^2/14400,
%F A371259 T(n, n-1) = 2^2 * Sum_{j=3..n-1} j^2.
%e A371259 Triangle begins:
%e A371259            1;
%e A371259           36,           1;
%e A371259         1764,         100,          1;
%e A371259       112896,        9864,        200,        1;
%e A371259      9144576,     1099296,      34064,      344,     1;
%e A371259    914457600,   142159392,    6004512,    92200,   540,  1;
%e A371259 110649369600, 21385410048, 1156921920, 24075712, 213700, 796, 1.
%e A371259   ...
%e A371259 An example for T(4, 3). The corresponding partitions are
%e A371259 pi(1) = {(1),(2),(3,4)},
%e A371259 pi(2) = {(1),(2),(4,3)},
%e A371259 pi(3) = {(1),(3),(2,4)},
%e A371259 pi(4) = {(1),(3),(4,2)},
%e A371259 pi(5) = {(1,4),(2),(3)},
%e A371259 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}.
%e A371259 Compute the number of ordered 2-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) = 6^2 = 36.
%p A371259 T:= proc(n, k) option remember; `if`(k<3 or k>n, 0,
%p A371259       `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^2*T(n-1, k)))
%p A371259     end:
%p A371259 seq(seq(T(n, k), k=3..n), n=3..10);
%t A371259 A371259[n_, k_] := A371259[n, k] = Which[n < k || k < 3, 0, n == k, 1, True, A371259[n-1, k-1] + (n+k-1)^2*A371259[n-1, k]];
%t A371259 Table[A371259[n, k], {n, 3, 10},{k, 3, n}] (* _Paolo Xausa_, Jun 11 2024 *)
%o A371259 (Python)
%o A371259 A371259 = lambda n, k: 0 if (k < 3 or k > n) else (1 if (n == 3 and k == 3) else (A371259(n-1, k-1) + ((n + k - 1)**2) * A371259(n-1, k)))
%o A371259 print([A371259(n, k) for n in range(3, 11) for k in range(3, n+1)])
%Y A371259 Cf. A143498, A371081.
%K A371259 nonn,tabl
%O A371259 3,2
%A A371259 _Aleks Zigon Tankosic_, Mar 16 2024
%E A371259 More terms from _Michel Marcus_, Jun 12 2025