Counting rhymes -- Catalan, Bell numbers

     In mathematics, the Catalan numbers (named for Belgian mathematician Eugène Charles Catalan, 1814–1894, and beginning with 1, 1, 2, 5, 14, 42, 132, 429, . . . ) and the Bell numbers (named for the Scottish mathematician Eric Temple Bell, 1883-1960, and beginning with 1, 1, 2, 5, 15, 52, 203, 877,  . . . ),  provide answers to a variety of mathematical counting-problems, including counting the number of rhyme schemes for stanzas of poetry.  In English, earliest classification of rhyme schemes dates back to George Puttenham and his treatise, The Arte of English Poesie (published around 1590). 

     The first two of the Catalan and Bell number sequences (1 and 1) refer to rhyme schemes for poems of zero lines and one line.  The 3rd Catalan-Bell number -- which is 2 -- gives the number of rhyme schemes in a poem of two lines:  either the two lines rhyme or they do not.  The 4th Catalan-Bell number -- which is 5 -- is the number of rhyme schemes for a stanza of 3 lines:  all three of the lines may rhyme, or none, or any two (three possibilities here: 12, 13, 23).

     For stanzas of 4 (or more) lines, diagrams are useful to keep track of the various rhyme schemes; I offer this table of  15 "Puttenham" diagrams -- in which rhyming lines in a 4-line-stanza are joined at the right-end by arcs -- as introduced in The Arte of English Poesie cited above.

Careful examination of this diagram shows that all possible rhyme schemes for a 4-line stanza are demonstrated here.  All together there are 15 possible rhyme schemes, and 14 of these have no lines crossing in the Puttenham diagram. These 14 rhyme schemes without crossings are counted by the 5th Catalan number -- and all possible rhyme schemes are tallied by the 5th Bell number, 15.

In summary, rhyme schemes for stanzas of 3 or fewer lines require no crossings in the Puttenham diagram -- for these, the Bell and Catalan numbers agree.  For larger stanzas, at least one of the diagrams involves crossings of rhyme-arcs and the Bell number is larger than the corresponding Catalan number.

Wikipedia and other online sites offer formulas for calculating Catalan and Bell numbers.

In keeping with our topic, we end with a small poem -- a traditional "counting rhyme":

     One, Two, Buckle My Shoe

     One, two, buckle my shoe;
     Three, four, knock at the door;
     Five, six, pick up sticks;
     Seven, eight, lay them straight;
     Nine, ten, a good fat Hen;
     Eleven, twelve, dig and delve;
     Thirteen, fourteen, maids a-courting;
     Fifteen, sixteen, maids a-kissing;
     Seventeen, eighteen, maids a-waiting;
     Nineteen, twenty, my plate's empty.