A371277 -id:A371277 - cp's OEIS frontend

A373761 Row sums of A371277.

1, 65, 8281, 1832833, 648383681, 344889060481, 263241070687225, 277821953546314241, 393388714018670749633, 728818437848717556976321, 1729501321168395811504313561, 5161574686582090379099550582145, 19067391184605659825359397658612481, 85976151321068604971683536102183592193

Offset: 2

Aleks Zigon Tankosic, Jun 17 2024

nonn

A. Žigon Tankosič, The (l,r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A371277.

a := proc(n) local T, k; T := proc(n, k) option remember; if n = k then 1; elif k < 2 or n < k then 0; else T(n - 1, k - 1) + (n + k - 1)^3*T(n - 1, k); end if; end proc; add(T(n, k), k = 2 .. n); end proc:
seq(a(n), n = 0 .. 18);

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

Original entry on oeis.org

1, 16, 1, 400, 52, 1, 14400, 2948, 116, 1, 705600, 203072, 12344, 216, 1, 45158400, 17154432, 1437472, 38480, 360, 1, 3657830400, 1760601600, 191088544, 6978592, 99320, 556, 1, 365783040000, 216690624000, 29277351936, 1370470592, 26445312, 224420, 812, 1

Offset: 2

Aleks Zigon Tankosic, Mar 10 2024

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).

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).

			Triangle begins:
         1;
        16,          1;
       400,         52,         1;
     14400,       2984,       116,       1;
    705600,     203072,     12344,     216,     1;
  45158400,    1715443,   1437472,   38480,   360,    1;
3657830400, 1760601600, 191088544,  6978592, 99320, 556, 1.
  ...
An example for T(4, 3). The corresponding partitions are
pi(1) = {(1),(2),(3,4)},
pi(2) = {(1),(2),(4,3)},
pi(3) = {(1),(2,3),(4)},
pi(4) = {(1),(3,2),(4)},
pi(5) = {(1),(2,4),(3)},
pi(6) = {(1),(4,2),(3)},
pi(7) = {(1,3),(2),(4)},
pi(8) = {(3,1),(2),(4)},
pi(9) = {(1,4),(2),(3)},
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
 bl(pi(3)) = bl(pi(4)) = bl(pi(7)) = bl(pi(8)) = {1,2,4}.
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.

A. Žigon Tankosič, The (l, r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Column k=2 gives A001715(n+1)^2.

Cf. A143497, A371259, A371277, A371488 (row sums), A372208.

T:= proc(n, k) option remember; `if`(k<2 or k>n, 0,
      `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^2*T(n-1, k)))
    end:
seq(seq(T(n, k), k=2..n), n=2..10);  # Alois P. Heinz, Mar 11 2024

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]];
Table[A371081[n, k], {n, 2, 10}, {k, 2, n}] (* Paolo Xausa, Jun 12 2024 *)

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)))
print([A371081(n, k) for n in range(2, 10) for k in range(2, n+1)])

Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^2*T(n-1, k).
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.
Special cases:
T(n, k) = 0 for n < k or k < 2,
T(n, n) = 1,
T(n, 2) = (A143497(n,2))^2 = A001715(n+1)^2 = ((n+1)!)^2/36,
T(n, n-1) = 2^2 * Sum_{j=2..n-1} j^2.

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

Original entry on oeis.org

1, 216, 1, 74088, 728, 1, 37933056, 604800, 1728, 1, 27653197824, 642733056, 2904768, 3456, 1, 27653197824000, 883130895360, 5662172160, 10497600, 6200, 1, 36806406303744000, 1553703385006080, 13322923130880, 34467586560, 31422600, 10296, 1, 63601470092869632000, 3450292743162101760, 38111804456140800, 129651027770880, 163174556160, 82006848, 16128, 1

Offset: 3

Aleks Zigon Tankosic, Apr 22 2024

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).

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).

			Triangle begins:
              1;
            216,             1;
          74088,           728,           1;
       37933056,        604800,        1728,         1;
    27653197824,     642733056,     2904768,      3456,     1;
 27653197824000,  883130895360,  5662172160,  10497600,  6200,  1.
  ...
An example for T(4, 3). The corresponding partitions are
pi(1) = {(1),(2),(3,4)},
pi(2) = {(1),(2),(4,3)},
pi(3) = {(1),(3),(2,4)},
pi(4) = {(1),(3),(4,2)},
pi(5) = {(1,4),(2),(3)},
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.

A. Žigon Tankosič, The (l,r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A143498, A371081, A371259, A371277.

T:= proc(n, k) option remember; `if`(k<3 or k>n, 0,
      `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^3*T(n-1, k)))
    end:
seq(seq(T(n, k), k=3..n), n=3..10);

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]];
Table[A372208[n, k], {n, 3, 10}, {k, 3, n}] (* Paolo Xausa, Jun 11 2024 *)

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)))
print([A372208(n, k) for n in range(3, 11) for k in range(3, n+1)])

Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^3*T(n-1, k).
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.
Special cases:
T(n, k) = 0 for n < k or k < 3, [corrected by Paolo Xausa, Jun 11 2024]
T(n, n) = 1,
T(n, 3) = (A143498(n, 3))^3 = ((n+2)!)^3/1728000,
T(n, n-1) = 2^3 * Sum_{j=3..n-1} j^3.

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A373761 Row sums of A371277.

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

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

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