342 



PROFESSOR TAIT ON KNOTS. 



Table of the values of p* ; the number of partitions of s in which no one is 

 less than 2, nor greater than r. 



{The values of v are in the first row, those of s in the first column.) 



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 















1 

























1 













. 



. 





















2 











1 



. 



. 





. 

















3 















1 







. 

















4 











1 



1 



2 





. 

















5 















1 



1 



2 



. 

















6 











1 



2 



3 



3 



4 

















7 















1 



2 



3 



3 



4 















8 











1 



2 



4 



5 



6 



6 



7 













9 















2 



3 



5 



6 



7 



7 



8 











10 











1 



2 



5 



7 



9 



10 



11 



11 



12 









11 















2 



4 



7 



9 



11 



12 



13 



13 



14 







12 











1 



3 



7 



10 



14 



16 



18 



19 



20 : 



20 : 



21 . 





13 















2 



5 



10 



13 



17 



19 



21 



22 



23 



23 24 . 





14 











1 



3 



8 



13 



19 



23 



27 



29 



3i : 



52 : 



33 33 34 . 





15 















3 



7 



14 



20 



26 



30 



34 



36 



38 



39 40 40 41 





16 











1 



3 



10 



17 



26 



33 



40 



44 



48 50 52 53 54 54 55 . 



17 















3 



8 



18 



27 



37 



44 



51 



55 59 61 63 64 65 65 6 



18 











1 



4 



12 



22 



36 



47 



58 



66 



73 77 81 83 85 86 87 . 



19 















3 



10 



23 



36 



52 



64 



75 



83 90 94 98 100 102 





20 











1 



4 



14 



28 



47 



64 



82 



95 



107 115 122 126 130 . 





21 















4 



12 



29 



49 



72 



91 



110 



123 135 143 150 . 





22 











1 



4 



16 



34 



60 



86 



113 



134 



154 168 180 . 





23 















4 



14 



36 



63 



96 



126 



155 



177 197 . 







24 











1 



5 



19 



42 



78 



115 



155 



189 



220 









25 















4 



16 



44 



80 



127 



171 



215 











26 











1 



5 



21 



50 



97 



149 



207 













27 















5 



19 



53 



102 



166 















28 











1 



5 



24 



60 



120 

















29 















5 



21 



63 



. 



. 















30 











1 



6 



27 



. 





. 





. 











31 















5 



. 



. 





. 





. 











32 











1 



• 



• 



• 



• 



• 





• 



• 









From what has been stated in the previous pages, it is easy to see how to 

 extend this table ; forming the successive terms of each row by adding step by 

 step upwards to the right along a diagonal, thence upwards to the top, zig-zag 

 along the row of heavier type as soon as it is reached. 



