By using symmetry, from a short symbol [`3],2,2 we obtain (3,2,2) and (2 1,2,2), from [`4],2,2 we obtain (4,2,2), (3 1,2,2), (2 2,2,2), (2 1 1,2,2), from [`3],[`3],2 we obtain (3,3,2), (3,2 1,2) = (2 1,3,2) and (2 1,2 1,2), and then eliminate duplications. Continuing in this way, we obtain all alternating KLs of the stellarrational subworld. Thus, the stellar world is completely included in the SRworld. Instead of repeating the complete derivation of S^{+} links from the source links of the type 2,2,...,2+k, (k = 1,2,...), they can be obtained directly from the stellarrational KLs derived, by adding an appropriate number of pluses. For n = 7 we add one plus to stellarrational KLs with 6 crossings, for n = 8 we derive S^{+} links from SRlinks with 6 crossings by adding two pluses, and from SRlinks with 7 crossings by adding one plus, etc. This way, for every n we derive KLs with k pluses from SRlinks with nk, ..., n2, n1 crossings (n ³k+6, k = 1,2,...). The LinKnot functions fStellarBasic (webMathematica fStellarBasic), fStellar (webMathematica fStellar), and fStellarPlus (webMathematica fStellarPlus) calculate the number and Conway symbols of all stellar and stellarrational KLs without and with pluses for a given number of crossings n, respectively. Here we encountered different combinatorial problems. For example, for every fixed n we could try to calculate the number of different classes in the first column of Table 1 (beginning with series 1, 1, 2, 3, 4, 5, 7 for n£ 12), or the number of source KLs, or even the number of all KLs derived from some generating KL for every n fixed, etc. For example, the number of different classes mentioned is equal to the coefficient corresponding to q^{n6} in , where is the Gauss polynomial
All such problems belong to the theory of partitions with a given symmetry group (Ppartitions). Let P be a permutation group P on k objects and n ³ k be an integer. To every object k_{i} (1 £k_{i}£k) a natural number n_{i} is assigned, where å_{i = 1}^{k} n_{i} = n. Two partitions defined by such signed (or weighted) objects are equal iff there is a permutation from P transforming one to another. We need to find and enumerate different Ppartitions. In some special cases, we can reduce problems of Ppartitions to classical partition theory, but in general, this enumeration is an open problem.
