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Dec
8
comment Counting path generating sentences in a specific formal language
For an arbitrary Turing machine there is little or nothing that can be said.
Dec
7
comment Nontrivial question about fibonacci numbers?
More generally, for $p\ne 2$ or 5, $F_{n+p} \equiv F_{n+1} \pmod p$ if $(p|5)=1$ and $F_{n+p}\equiv -F_{n-1} \pmod p$ if $(p|5) = -1$. (Here $F_{-1}=1$.)
Dec
5
comment A natural sum over multisets (expectation over multinomial)
The generating function $f(x)=\sum_n n^n x^n/n!$ is equal to $1/(1+W(-x))$, where $W$ is the Lambert $W$-function. The denominator has a zero at $x=1/e$ which should enable you to get a good approximation to the coefficients of $f(x)^k$ by standard techniques (see, e.g., Flajolet and Sedgewick's Analytic Combinatorics).
Nov
30
revised Dyck paths on rectangles
improved TeX formatting
Nov
29
comment Number of Dyck paths with k returns and b peaks
See Emeric Deutsch, Dyck Path Enumeration, Discrete Mathematics 204 (1999) 167-202, section 6.5, sciencedirect.com/science/article/pii/S0012365X98003719
Nov
28
comment Inclusion-preserving bijection between subsets of cardinality k and n-k
See also Curtis Greene and Daniel J. Kleitman. Strong versions of Sperner’s theorem. J. Combinatorial Theory Ser. A, 20(1):80–88, 1976. For similar decompositions of other posets, search for "symmetric chain decomposition".
Nov
11
awarded  Yearling
Nov
2
revised Combinatorial Morse functions and random permutations
added 138 characters in body
Nov
1
comment Where can I buy Napier's Bones/Rods?
I put it in back quotes which works, but puts it in a gray box which I couldn't get rid of.
Nov
1
revised Where can I buy Napier's Bones/Rods?
added 6 characters in body
Oct
31
comment What are some examples of interesting uses of the theory of combinatorial species?
OK, Joyal described his proof using species, but he used (implicitly) a non-natural bijection; more precisely, he used the fact that the number of linear orders of a finite set is equal to the number of permutations (i.e., sets of cycles) of the set, but the corresponding species are not isomorphic. The proof may have been inspired by species, but I don't think it's a good example of what species are good for. There are other ways to deal with exponential generating functions; the real power of species (at least in enumeration) is its application in enumeration under group action.
Oct
30
answered What are some examples of interesting uses of the theory of combinatorial species?
Oct
30
comment What are some examples of interesting uses of the theory of combinatorial species?
But Joyal's proof does not use species. His proof may have been inspired by species, but his bijection is not natural (i.e., not functorial) — not that there's anything wrong with that.
Oct
24
comment Finite series with reciprocal factorials
This hypergeometric expression says no more and no less than that the coefficient of $x^{k+2}$ in the sum is $$\frac{1}{(n-k)!\,(k+2)}.$$
Oct
24
awarded  Enlightened
Oct
24
awarded  Nice Answer
Oct
22
comment Representing numbers in a non-integer base with few (but possibly negative) nonzero digits
Alpert's theorem was proved earlier by Clemens Heuberger, "Minimal expansions in redundant number systems: Fibonacci bases and greedy algorithms", Periodica Mathematica Hungarica 49 (2), 2004, 65–89.
Oct
21
revised A number array related to colored necklaces and the primes
added 39 characters in body
Oct
21
revised A number array related to colored necklaces and the primes
edited body
Oct
21
revised A number array related to colored necklaces and the primes
added 3 characters in body