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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 $$\frac{1}{(nk)!\,(k+2)}.$$ 
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Oct 22 
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Representing numbers in a noninteger 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 22 
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KPermutations with forbidden numbers
Search for "permutations with forbidden positions" and "rook polynomials". 
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 
Oct 21 
revised 
A number array related to colored necklaces and the primes
added 246 characters in body 
Oct 21 
answered  A number array related to colored necklaces and the primes 
Oct 15 
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A generalisation of Narayanalike numbers (walks on the 2D lattice)
See mat.univie.ac.at/~kratt/artikel/encystat.pdf. A similar question is mathoverflow.net/questions/164389/numberofwalks. 
Oct 13 
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Name for series $\sum f_n x^n / (n! (n+k)!)$
It's unlikely that these will have any combinatorial applications since the product of two "Bessel generating functions" with integer coefficients won't in general have integer coefficients (by which I mean the coefficients of $x^n/n!(n+k)!$). 
Oct 13 
answered  Name for series $\sum f_n x^n / (n! (n+k)!)$ 
Oct 7 
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Counting Problems where Labeled is Known but Unlabeled is Not
I would say that a lot is known about the count of unlabeled trees; it's just more complicated than for labeled trees, and this is very typical of graphical enumeration. You can find lots of examples in Harary and Palmer's book "Graphical Enumeration." There is even an explicit though complicated formula (as a sum over partitions) for the number of unlabeled trees. It might be noted that there are some problems, such as counting selfcomplementary graphs, in which the unlabeled version has been solved, but not the labeled version. 
Oct 7 
answered  Is there a formula for the number of labeled forests with $k$ components on $n$ vertices? 
Oct 2 
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Is there a bijection of permutations onto mathematical objects that preserve information about descents?
The RobinsonSchensted correspondence restricted to FPFI is a bijection onto standard Young tableaux with every column of even length that preserves descents (i is a descent of a standard Young tableau if i+1 is to the left of i). 
Sep 30 
awarded  Explainer 
Aug 11 
awarded  Nice Answer 
Aug 11 
revised 
Combinatorial Morse functions and random permutations
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Jul 23 
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Combinatorial Proof of Real Analysis Identity
I believe that this identity is a limit of a finite sum that can be proved using the WZ method. Whether that makes it combinatorial is a matter of opinion. 