| Function Abstraction | Symbol | AN⁴ | Bird | CR¹ | CH² | Elem³ | Haskell | APL | BQN | Intro |
| λa.a | I | - | Identity | SKK | SKK | ✅ | id | ⊣⊢ | ⊣⊢ | Sch24 |
| λab.a | K | - | Kestrel | K | K | ✅ | const | ⊢ | ⊢ | Sch24 |
| λab.b | κ | - | Kite | KI | KI | ⊣ | ⊣ | |||
| λab.abb | W | 1y2 | Warbler | C(BMR) | CSK | ✅ | join | ⍨ | ˜ | Cur29 |
| λabc.acb | C | 2y2 | 🐦 Cardinal | S(BBS)(KK) | B(ΦBS)KK | ✅ | flip | ⍨ | ˜ | Sch24 |
| λabc.a(bc) | B | 1y11 | Bluebird | S(KS)K | S(KS)K | ✅ | . | ∘⍤⍥ 2T | ∘○ 2T | Sch24 |
| λabcd.a(bcd) | B₁ | 2y12 | Blackbird | BBB | DB | .: | ⍤ 2T | ∘ 2T | Cur30 | |
| λabcde.a(bcde) | B2 | 3y13 | Bunting | B(BBB)B | DB₁ | Cur30 | ||||
| λabcd.a(b(cd)) | B3 | 4y14 | Becard | B(BB)B | BDB | Cur30 | ||||
| λabc.ac(bc) | S | 1y21 | Starling | S | S | <*> / ap | ⟜ | Sch24 | ||
| λabc.a(bc)c | Σ | 1y12 | Violet Starling | - | ? | =<< | ⊸ | |||
| λabcd.ab(cd) | D | 2y21 | Dove | BB | BB | ∘ | ⟜ | Smu85 | ||
| λabcd.a(bc)d | Δ | 2y12 | Zebra Dove | - | ? | ⊸ | ||||
| λabcd.a(bd)(cd) | Φ | 1y121 | Phoenix | - | B₁SB | liftA2 | 3T | 3T | Cur31 | |
| λabcd.a(bc)(bd) | Ψ | 2y21 | Psi | - | B(SΦCB)B | on | ⍥ | ○ | Cur31 | |
| λabcde.abc(de) | D1 | 3y31 | Dickcissel | B(BB) | BD | |||||
| λabcde.a(bc)(de) | D2 | 2y211 | Dovekie | BB(BB) | DD | Smu85 | ||||
| λabcde.ab(cde) | E | 3y22 | 🦅 Eagle | B(BBB) | BB₁ | Smu85 | ||||
| λabcde.a(bcd)e | ε | 3y22 | Golden Eagle | - | ? | |||||
| λabcde.a(bde)(cde) | Φ1 | 2y222 | Pheasant | - | BΦΦ | 3T | 3T | Cur31 | ||
| λabcdefg.a(bcd)(efg) | Ê | 4y222 | Bald Eagle | B(BBB)(B(BBB)) | D₂D₂D | Smu85 |
1 - CR = Chris Rathman (combinator spellings taken from his site)
2 - CH = Conor Hoekstra (combinator spellings generated from C++ program)
3 - Elementary combinators are listed in Chapter 5 of Combinatory Logic: Volume I
4 - Arity Notation: AyBC = An A-arity yielding binary function (because BC is length two) where the arguments of the function have arity B and C
| Author | Year | Paper | |
| Sch24 | Moses Schönfinkel | 1924 | On the building blocks of mathematical logic |
| Cur29 | Haskell Curry | 1929 | An Analysis of Logical Substitution |
| Cur30 | Haskell Curry | 1931 | Grundlagen der Kombinatorischen Logik (The Foundations of Combinatory Logic) |
| Cur31 | Haskell Curry | 1931 | The universal quantifier in combinatory logic |
| Cur48 | Haskell Curry | 1948 | A Simplification of the Theory of Combinators |
| Cur58 | H. Curry & R. Feys | 1958 | Combinatory Logic: Volume I |
| Tur78 | David Turner | 1979 | Another algorithm for bracket abstraction |
| Smu85 | Raymond Smullyan | 1985 | To Mock a Mockingbird |
| Iv89 | K. Iverson & E. McDonnell | 1989 | Phrasal Forms |
| Loc12 | Marshall Lochbaum | 2012 | Added hook, backhook |