A082641 - OEIS

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.

%I #28 Nov 09 2018 20:27:46

%S 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,

%T 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,

%U 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

%N 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.

%C 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

%D M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 208.

%D C. W. Strom, Complete systems of invariants of the cyclic groups of equal order and degree, Proc. Iowa Acad. Sci., 55 (1948), 287-290.

%H Finklea, Moore, Ponomarenko and Turner, <a href="http://www-rohan.sdsu.edu/~vadim/fmpt1b-revised.pdf">Invariant Polynomials and Minimal Zero Sequences</a>, Involve 1 (2008), no. 2, 159-165.

%H Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner, <a href="http://dx.doi.org/10.2140/involve.2008.1.159">Invariant polynomials and minimal zero sequences</a>, Involve, 1:2 (2008), pp. 159-165.

%H Vadim Ponomarenko, <a href="http://www-rohan.sdsu.edu/~vadim/Cyclic.xls">Table</a> (Excel spread-sheet format)

%H Vadim Ponomarenko, <a href="http://www-rohan.sdsu.edu/~vadim/mzs.zip">Programs</a>

%e Triangle with row sums (A002956):

%e Z_1: 1 ................................... 1

%e Z_2: 1 1 ................................ 2

%e Z_3: 1 1 2 ............................. 4

%e Z_4: 1 2 2 2 .......................... 7

%e Z_5: 1 2 4 4 4 ...................... 15

%e Z_6: 1 3 6 6 2 2 ................... 20

%e Z_7: 1 3 8 12 12 6 6 ................ 48

%e Z_8: 1 4 10 18 16 8 4 4 ............. 65

%e Z_9: 1 4 14 26 32 18 12 6 6 ......... 119

%e Z_10: 1 5 16 36 48 32 12 8 4 4 ...... 166

%e Z_11: 1 5 20 50 82 70 50 30 20 10 10 ... 348

%e ...

%Y Row sums give A002956.

%K nonn,tabl

%O 1,6

%A _N. J. A. Sloane_, May 15 2003

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