A082641 Triangle T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = number of basic invariants of degree k for the cyclic group of order and degree n.

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 4, 1, 3, 6, 6, 2, 2, 1, 3, 8, 12, 12, 6, 6, 1, 4, 10, 18, 16, 8, 4, 4, 1, 4, 14, 26, 32, 18, 12, 6, 6, 1, 5, 16, 36, 48, 32, 12, 8, 4, 4, 1, 5, 20, 50, 82, 70, 50, 30, 20, 10, 10, 1, 6, 24, 64, 104, 84, 36, 20, 12, 8, 4, 4, 1, 6, 28, 84, 168, 180, 132, 84, 60, 36, 24, 12, 12, 1, 7, 32, 104, 216, 242, 162, 96, 42, 30, 18, 12, 6, 6

Offset: 1

Author

N. J. A. Sloane, May 15 2003

Keywords

• nonn
• tabl

Comments

T(n,k) is also the number of multisets of k integers ranging from 1 to n, such that the sum of members of the multiset is congruent to 0 mod n, and no submultiset exists whose sum of members is congruent to 0 mod n. - Andrew Weimholt, Jan 31 2011

Examples

			Triangle with row sums (A002956):
  Z_1:  1  ................................... 1
  Z_2:  1  1  ................................ 2
  Z_3:  1  1  2  ............................. 4
  Z_4:  1  2  2  2  .......................... 7
  Z_5:  1  2  4  4  4  ...................... 15
  Z_6:  1  3  6  6  2  2  ................... 20
  Z_7:  1  3  8 12 12  6  6  ................ 48
  Z_8:  1  4 10 18 16  8  4  4  ............. 65
  Z_9:  1  4 14 26 32 18 12  6  6  ......... 119
  Z_10: 1  5 16 36 48 32 12  8  4  4  ...... 166
  Z_11: 1  5 20 50 82 70 50 30 20 10 10  ... 348
  ...
		

References

• M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 208.
• C. W. Strom, Complete systems of invariants of the cyclic groups of equal order and degree, Proc. Iowa Acad. Sci., 55 (1948), 287-290.

Links

• Finklea, Moore, Ponomarenko and Turner, Invariant Polynomials and Minimal Zero Sequences, Involve 1 (2008), no. 2, 159-165.
• Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner, Invariant polynomials and minimal zero sequences, Involve, 1:2 (2008), pp. 159-165.
• Vadim Ponomarenko, Table (Excel spread-sheet format)
• Vadim Ponomarenko, Programs

Crossrefs

Row sums give A002956.

Extensions

More terms from Vadim Ponomarenko (vadim123(AT)gmail.com), Jun 29 2004