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visits | member for | 4 years, 1 month |
seen | 20 mins ago | |
stats | profile views | 1,013 |
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 |
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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 |
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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 |
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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 |