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A371081 Triangle read by rows, (2, 2)-Lah numbers.

%I A371081 #87 Aug 17 2025 11:18:14
%S A371081 1,16,1,400,52,1,14400,2948,116,1,705600,203072,12344,216,1,45158400,
%T A371081 17154432,1437472,38480,360,1,3657830400,1760601600,191088544,6978592,
%U A371081 99320,556,1,365783040000,216690624000,29277351936,1370470592,26445312,224420,812,1
%N A371081 Triangle read by rows, (2, 2)-Lah numbers.
%C A371081 The (2, 2)-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 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 A371081 The (2, 2)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=r=2. 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 A371081 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 A371081 Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^2*T(n-1, k).
%F A371081 Explicit formula: T(n, k) = Sum_{3 <= j(1) < j(2) < ... < j(n-k) <= n} (2j(1)-2)^2 * (2j(2)-3)^2 * ... * (2j(n-k)-(n-k+1))^2.
%F A371081 Special cases:
%F A371081 T(n, k) = 0 for n < k or k < 2,
%F A371081 T(n, n) = 1,
%F A371081 T(n, 2) = (A143497(n,2))^2 = A001715(n+1)^2 = ((n+1)!)^2/36,
%F A371081 T(n, n-1) = 2^2 * Sum_{j=2..n-1} j^2.
%e A371081 Triangle begins:
%e A371081          1;
%e A371081         16,          1;
%e A371081        400,         52,         1;
%e A371081      14400,       2984,       116,       1;
%e A371081     705600,     203072,     12344,     216,     1;
%e A371081   45158400,    1715443,   1437472,   38480,   360,    1;
%e A371081 3657830400, 1760601600, 191088544,  6978592, 99320, 556, 1.
%e A371081   ...
%e A371081 An example for T(4, 3). The corresponding partitions are
%e A371081 pi(1) = {(1),(2),(3,4)},
%e A371081 pi(2) = {(1),(2),(4,3)},
%e A371081 pi(3) = {(1),(2,3),(4)},
%e A371081 pi(4) = {(1),(3,2),(4)},
%e A371081 pi(5) = {(1),(2,4),(3)},
%e A371081 pi(6) = {(1),(4,2),(3)},
%e A371081 pi(7) = {(1,3),(2),(4)},
%e A371081 pi(8) = {(3,1),(2),(4)},
%e A371081 pi(9) = {(1,4),(2),(3)},
%e A371081 pi(10) = {(4,1),(2),(3)}, since A143497 for n=4, k=3 equals 10. Sets of their block leaders are bl(pi(1)) = bl(pi(2)) = bl(pi(5)) = bl(pi(6)) = bl(pi(9)) = bl(pi(10)) = {1,2,3} and
%e A371081  bl(pi(3)) = bl(pi(4)) = bl(pi(7)) = bl(pi(8)) = {1,2,4}.
%e A371081 Compute the number of ordered 2-tuples (i.e., ordered pairs) of partitions pi(1), pi(2), ..., pi(10) 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}, and four partitions with the set of block leaders equal to {1,2,4}, T(4, 3) = 6^2 + 4^2 = 52.
%p A371081 T:= proc(n, k) option remember; `if`(k<2 or k>n, 0,
%p A371081       `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^2*T(n-1, k)))
%p A371081     end:
%p A371081 seq(seq(T(n, k), k=2..n), n=2..10);  # _Alois P. Heinz_, Mar 11 2024
%t A371081 A371081[n_, k_] := A371081[n, k] = Which[n < k || k < 2, 0, n == k, 1, True, A371081[n-1, k-1] + (n+k-1)^2*A371081[n-1, k]];
%t A371081 Table[A371081[n, k], {n, 2, 10}, {k, 2, n}] (* _Paolo Xausa_, Jun 12 2024 *)
%o A371081 (Python)
%o A371081 A371081 = lambda n, k: 0 if (k < 2 or k > n) else (1 if (n == 2 and k == 2) else (A371081(n-1, k-1) + ((n + k - 1)**2) * A371081(n-1, k)))
%o A371081 print([A371081(n, k) for n in range(2, 10) for k in range(2, n+1)])
%Y A371081 Column k=2 gives A001715(n+1)^2.
%Y A371081 Cf. A143497, A371259, A371277, A371488 (row sums), A372208.
%K A371081 nonn,tabl
%O A371081 2,2
%A A371081 _Aleks Zigon Tankosic_, Mar 10 2024