A new paradigm for designing ring construction strategies for green organic synthesis: implications for the discovery of multicomponent reactions to build molecules containing a single ring

A new way of developing novel synthesis strategies for the construction of monocyclic rings found in organic molecules is presented. The method is based on the visual application of integer partitioning to chemical structures. Two problems are addressed: (1) the determination of the total number of possible ways to construct a given ring by 2-, 3-, and 4-component couplings; and (2) the systematic enumeration of those possibilities. The results of the method are illustrated using cyclohexanone, pyrazole, and the Biginelli adduct as target ring systems with a view to discover new and greener strategies for their construction using multicomponent reactions. The application of the method is also extended to various heterocycles found in many natural products and pharmaceuticals.

. Ladder pattern for determining the total number of 3-partitions of evenmembered monocyclic rings Table S2. Ladder pattern for determining the total number of 3-partitions of oddmembered monocyclic rings Table S3. Ladder pattern for determining the total number of 4-partitions of evenmembered monocyclic rings Table S4. Ladder pattern for determining the total number of 4-partitions of oddmembered monocyclic rings Enumeration of 3-partitions of monocyclic rings Enumeration of 4-partitions of monocyclic rings Schemes S1 to S3 showing three-component coupling strategies to synthesize cyclohexanone. Figure S1. Nucleophilic-electrophilic labelling of centres in 3-partition fragments of cyclohexanone. Figure S2. Nucleophilic-electrophilic labelling of centres in 2-partition fragments of piperdine. Figure S3. Nucleophilic-electrophilic labelling of centres in 3-partition fragments of piperdine. Figure S4. Nucleophilic-electrophilic labelling of centres in 2-partition fragments of cyclopentanone. Figure S5. Nucleophilic-electrophilic labelling of centres in 3-partition fragments of cyclopentanone. Figure S6. Nucleophilic-electrophilic labelling of centres in 2-partition fragments of pyrrolidine. Figure S7. Nucleophilic-electrophilic labelling of centres in 3-partition fragments of pyrrolidine.
Schemes S4 to S5 showing new three-component coupling strategies to synthesize the Biginelli adduct.
Note that for r = 3 the sequence term is 1.

Enumeration of 3-partitions of monocyclic rings:
Step 1: For a given ring size begin with a horizontal list of 2-partitions (n, m), where n is always larger than m.
Step 2. Under each (n, m), write out all 2-partitions of n in descending order in a column as follows: (k, l), m, where k > l.
Step 3: The unique 3-partitions in the array are given by (k, l, m) such that k > l > m.
Example 1: For a ring size of 12 we have the following array.
Note that partitions of the form  , , , , , , , , , , , , , , and , , a b c b c a c a b c b a a c b b a c are equivalent due to the inherent cyclic nature of the ring read in clockwise and anticlockwise senses. For example, (9,2,1) is equivalent to (9,1,2).

Example 2:
For a ring size of 11 we have the following array.
Even-membered rings will always terminate in a 2-partition equal to , 22 Odd-membered rings will always terminate in a 2-partition equal to 11 , 22 rr     .
If r is even and divisible by 3, then

Enumeration of 4-partitions of monocyclic rings:
Step 1: For a given ring size begin with a horizontal list of 3-partitions (n, m, l), where n > m > l as determined by the method of enumeration of 3-partitions described above.
Step 2. Under each (n, m, l), write out all 2-partitions of n in descending order in a column as follows: (u, v), m, l where u > v.
Step 3: Repeat step 2 for the m values.
Step 4: Select unique 4-partitions from array. For 4-partitions containing two identical digits, such as (x, x, y, z), ensure that the form (x, y, x, z) is also present in the unique set.

Example 1:
For a ring size of 12 we have the following array.

Example 2:
For a ring size of 11 we have the following array.