“Etaoin shrdlu” is a term from the era of typesetting with hot type, representing the 12 most frequently used letters in the English language. (For more background, see https://en.wikipedia.org/wiki/Etaoin_shrdlu)

Several places (including, for example, https://digg.com/2018/two-letter-initials-how-common), note that among people’s names in the United States, the most common first initials are C, D, J, M, and R. The most common initials of last names are B, M, S and W. And the most-likely two letter combination is JB. At the other end of that spectrum, least common initials (for both first and last names) are Q, X, Y, and Z.

This got me thinking about the 44 people who have been president, and the 48 who have been vice president. Among both presidents and vice presidents, the most common first initial is J. There have been ten presidents and six vice presidents whose names started with J. Among the presidents, there were six James (Madison, Monroe, Polk, Buchanan, Garfield, and Carter) and four Johns (both Adams, Tyler, and Kennedy). Well, actually five Johns, but John Calvin Coolidge used his middle name. Among the vice presidents, there were five Johns (Adams, Calhoun, Tyler, Breckinridge, and Garner) and one James (Sherman). Coolidge was also a vice president, giving us another unused John, and Vice President Dan Quayle’s full name is James Danforth Quayle.

Among presidents, the next most common first initials are W and G (six and five, respectively). But from that list of common initials? C, D, M, and R: two, two, two, and three presidents. For vice presidents, five each used the first initials A, C, G, and H. There were one D, three Ms, and three Rs.
It’s when we get to last names that we see a significant diversion from the population at large (remember, most common are B, M, S, and W). The most common last initials among presidents are both H and T (five of each). There have been three Bs, three Ms, and two Ws, but absolutely no presidents whose last names start with S. Among vice presidents, C and B are the most popular (six C and five B), with three Ms, two Ss, and three Ws.

To date, we have had no presidents whose last names started with D, I, Q, S, U, X, Y, or Z. And their absent first initials are: E, I, K, N, O, P, Q, S, V, X, and Y. So it’s definitely time for a President Ian Strock, right?
Among vice presidents, none have had last names starting with E, I, L, O, U, X, Y, or Z. Absent vice presidential first initials: B, F, I, K, O, P, Q, U, V, X, Y, or Z.

Unique first initials: Elbridge (Gerry), Nelson (Rockefeller), Ulysses (Grant, though his birth name was Hiram), and Zachary (Taylor).
Unique last initials: (Dwight) Eisenhower, (Abraham) Lincoln, (Dan) Quayle, and (Barack) Obama. And sort of (Richard) Nixon and (Martin) Van Buren, since they’re each the only one with their last initial, but they appear on both the presidents and vice presidents lists.

With so many people sharing so few initials, it ought to have been common for presidents and vice presidents to share initials, right? Wrong. The only time our president and vice president shared initials was when James Buchanan and John C. Breckinridge were in office (1857-61).
Another feature that caught my eye were alliterative initials. In fiction, they’re common: Bilbo Baggins, Betty Boop, the Bunnys (Babs, Bugs, and Buster), Cliff Clavin, the Duck (Daffy, Daisy, and Donald), Fred Flintstone, Henry Higgins, Humbert Humbert, King Kong, Lana Lang and Lois Lane, the Mouses (Mickey, Mighty, and Minnie), Olive Oyl, the Pink Panther, the Pigs (Petunia and Porky), Roger Rabbit, Tiny Tim, Willie Wonka, many characters who live in the Marvel Universe*, half the supporting cast of the Harry Potter series**… and it’s not unnoticed in the realm of presidents or vice presidents, either. We’ve had presidents Woodrow Wilson (1913-21, though his unused first name was actually Thomas), Calvin Coolidge (1923-29, see above, John), Herbert Hoover (1929-33), and Ronald Reagan (1981-89). Vice Presidents: Hannibal Hamlin (1861-65), William Wheeler (1877-81), Calvin Coolidge (1921-23, again, John), Charles Curtis (1929-33), and Hubert Humphrey (1965-69).

And then there were those who were sequentially named. We had presidents James Madison (1809-17) and James Monroe (1817-25). And vice presidents Calvin Coolidge (1921-23), Charles Dawes (1925-29), and Charles Curtis (1929-33).
As for the current election, Donald Trump is the second president named with a D (after Dwight Eisenhower), and Mike Pence is the first vice president with a P on his last name. Joe Biden (already on the vice presidents’ list) would be the eleventh J president, and Kamala Harris would be the first K vice president.

* : In The Big Bang Theory episode “The Excelsior Acquisition” (season 3, episode 16), Raj lists Stan Lee’s alliterative character names, including: Bruce Banner, Reed Richards, Sue Storm, Steven Strange, Otto Octavius, Silver Surfer, Peter Parker, J. Jonah Jameson Junior, Dum Dum Dugan, Green Goblin, Matt Murdock, Pepper Potts, Victor Von Doom, Millie the Model, Fantastic Four, Daredevil, Invincible Iron Man, Happy Hogan, Curt Connors, and Fin Fang Foom. (See https://bigbangtheory.fandom.com/wiki/The_Excelsior_Acquisition)
** : First appearing in Harry Potter and the Philosopher’s Stone: Bathilda Bagshot, Bertie Bott, Daedalus Diggle, Dudley Dursley, Filius Flitwick, Gregory Goyle, Gellert Grindelwald, Morag MacDougal, Minerva McGonagall, Marlene McKinnon, Pansy Parkinson, Padma Patil, Parvati Patil, Piers Polkiss, Poppy Pomfrey, Quirinus Quirrell, Severus Snape, Vindictus Viridian, and William “Bill” Weasley.
First appearing in Harry Potter and the Chamber of Secrets: Colin Creevey, Godric Gryffindor, Gladys Gudgeon, Helga Hufflepuff, Martin Miggs, Rowena Ravenclaw, and Salazar Slytherin.
First appearing in Harry Potter and the Prisoner of Azkaban: Cho Chang, Florean Fortescue, Peter Pettigrew, and Stan Shunpike.
First appearing in Harry Potter and the Goblet of Fire: Broderick Bode and Joey Jenkins.
First appearing in Harry Potter and the Order of the Phoenix: Dilys Derwent, Inigo Imago, Luna Lovegood, and Willy Widdershins.
First appearing in Harry Potter and the Deathly Hallows: Arkie Alderton, Betty Braithwaite, Mary MacDonald, Ted Tonks, and Wendell Wilkins.
(See https://scifi.stackexchange.com/questions/98752/harry-potter-characters-with-alliterative-names)
See also chapter 7 in my The Presidential Book of Lists, and chapter 5 in my Ranking the Vice Presidents.