A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1

Offset

0,9

Comments

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

References

G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Links

Pontus von Brömssen, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Formula

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.

Inverse binomial transforms of the columns is given by triangle A096806.

Example

n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

Mathematica

trans[x_]:=If[x=={}, {}, Transpose[x]];
levptns[n_, k_]:=If[k==1, IntegerPartitions[n], Join@@Table[Select[Tuples[levptns[#, k-1]&/@y], And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#, ConstantArray[n, k-1]]&/@#)])&], {y, IntegerPartitions[n]}]];
Table[If[sum==k, 1, Length[levptns[k, sum-k]]], {sum, 0, 10}, {k, 0, sum}] (* Gus Wiseman, Jan 27 2019 *)

Crossrefs

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).

Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).

Cf. A096651, A096752, A096753.

Cf. A096806.

Cf. A042984.

Cf. A144150, A213427, A290353, A323718, A323719.

Sequence in context: A177767 A047030 A047120 * A293551 A099233 A303912

Adjacent sequences: A096748 A096749 A096750 * A096752 A096753 A096754

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 07 2004

Status

approved