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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,6
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
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
EXAMPLE
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
...
CROSSREFS
Row sums give A002956.
Sequence in context: A118821 A118824 A209402 * A334507 A342462 A239140
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, May 15 2003
EXTENSIONS
More terms from Vadim Ponomarenko (vadim123(AT)gmail.com), Jun 29 2004
STATUS
approved

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Last modified March 2 16:48 EST 2024. Contains 370497 sequences. (Running on oeis4.)