A096651 Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 3, 1, 1, 0, 1, 3, 1, 4, 1, 1, 0, 1, -1, 7, 1, 5, 1, 1, 0, 1, 15, -17, 14, 1, 6, 1, 1, 0, 1, -78, 133, -61, 25, 1, 7, 1, 1, 0, 1, 632, -1020, 529, -152, 41, 1, 8, 1, 1, 0, 1, -6049, 9826, -4989, 1506, -314, 63, 1, 9, 1, 1, 0, 1, 68036, -110514, 56161, -16668, 3532, -576, 92, 1, 10, 1, 1, 0, 1, -878337, 1427046, -724881, 214528, -44703, 7276, -972, 129, 1, 11, 1, 1, 0, 1, 12817659, -20827070, 10576885, -3123249, 647092, -103476, 13644, -1541, 175, 1, 12, 1, 1

Offset: 0

Author

Paul D. Hanna and Wouter Meeussen, Jul 02 2004

Keywords

• nice
• sign
• tabl

Comments

Hanna's Triangle: There exists a unique lower triangular matrix T, with ones on its diagonal, such that the row sums of T^n yields the n-dimensional partitions for all n>0. Specifically, row sums of T form A000041 (linear partitions); row sums of T^2 form A000219 (planar partitions); row sums of T^3 form A000293 (solid partitions); row sums of T^4 form A000334(4-D); row sums of T^5 form A000390(5-D); row sums of T^6 form A000416(6-D); row sums of T^7 form A000427(7-D). Rows indexed 9-13 were calculated by Wouter Meeussen.

Existence and integrality of Hanna's triangle has been proved in arXiv:1203.4419. (Suresh Govindarajan)

Examples

			Triangle T begins:
  {1},
  {0,1},
  {0,1,1},
  {0,1,1,1},
  {0,1,2,1,1},
  {0,1,1,3,1,1},
  {0,1,3,1,4,1,1},
  {0,1,-1,7,1,5,1,1},
  {0,1,15,-17,14,1,6,1,1},
  {0,1,-78,133,-61,25,1,7,1,1},
  {0,1,632,-1020,529,-152,41,1,8,1,1},
  {0,1,-6049,9826,-4989,1506,-314,63,1,9,1,1},
  {0,1,68036,-110514,56161,-16668,3532,-576,92,1,10,1,1},
  {0,1,-878337,1427046,-724881,214528,-44703,7276,-972,129,1,11,1,1},
  ...
  with row sums: {1,1,2,3,5,7,11,15,22,...} (A000041).
T^2 begins:
  {1},
  {0,1},
  {0,2,1},
  {0,3,2,1},
  {0,5,5,2,1},
  {0,7,7,7,2,1},
  {0,11,16,9,9,2,1},
  {0,15,15,31,11,11,2,1},
  {0,22,59,-4,54,13,13,2,1},
  ...
  with row sums: {1,1,3,6,13,24,48,86,...} (A000219).
		

Links

• S. Govindarajan Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.
• Wouter Meeussen, Rows 14-17 added

Crossrefs

Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096652(T^2), A096653(T^3), A096642-A096645(columns).

Cf. A096800, A096751.

Formula

For n>=0: T(0, 0)=1, T(n+1,0)=0, T(n+1,1)=1. For n>=1: T(n, n)=1, T(n+1, n)=1, T(n+2, n)=n, T(n+3, n)=1, T(n+4, n)=n*(5+n^2)/6, T(n+5, n)=(-48+90*n-7*n^2-6*n^3-5*n^4)/24, T(n+6, n)=(400-382*n-55*n^2+30*n^3+35*n^4+12*n^5)/40 (Wouter Meeussen). Corrected entry for the zeroth and first columns of the matrix T -- entry had columns and rows interchanged (Corrected by Suresh Govindarajan)

G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], where P_n(y) is the n-th row polynomial of triangle A096800.

Extensions

Rows 14-17 calculated (using extra terms in A096642-A096645 provided by Sean A. Irvine) by Wouter Meeussen, Jan 08 2011