A353132 Triangle read by rows of partial Bell polynomials B_{n,k}(x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.

2, 1, 4, 2, 6, 8, 3, 18, 24, 16, 4, 40, 100, 80, 32, 5, 78, 305, 440, 240, 64, 6, 140, 798, 1750, 1680, 672, 128, 7, 236, 1876, 5838, 8400, 5824, 1792, 256, 8, 378, 4056, 17136, 34524, 35616, 18816, 4608, 512, 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024

Offset

1,1

Links

Jordan Weaver, Rows 1 to 40 of triangle, flattened

E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.

E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.

Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.

A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.

A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Formula

T(n,k) = A353131(n,k)/(n-k+1)!

Sum_{k=1..n} T(n,k) = A349458(n).

Example

For n = 4, k = 2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3) = 4*x_1*x_3 + 3*x_2^2, so T(4,2) = B_{4,2}(2,2,12) - (4*2*12 + 3*2^2)/3! = 18.
Triangle begins:
   [1] 2;
   [2] 1,   4;
   [3] 2,   6,    8;
   [4] 3,  18,   24,    16;
   [5] 4,  40,  100,    80,     32;
   [6] 5,  78,  305,   440,    240,     64;
   [7] 6, 140,  798,  1750,   1680,    672,    128;
   [8] 7, 236, 1876,  5838,   8400,   5824,   1792,   256;
   [9] 8, 378, 4056, 17136,  34524,  35616,  18816,  4608,   512;
  [10] 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024.

Crossrefs

Cf. A000079, A353131, A349413, A268441, A178867.

Sequence in context: A239960 A330001 A292402 * A205685 A334404 A143375

Adjacent sequences: A353129 A353130 A353131 * A353133 A353134 A353135

Keyword

nonn,tabl

Author

Jordan Weaver, Apr 24 2022

Status

approved