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15h

awarded  Nice Answer 
1d

answered  Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle  is there a combinatorial explanation? 
May 22 
comment 
How long can a cycle of antichains in a finite partial order be?
Readers of my previous comment should realize that what I call $L_X$ has been changed in the question to $Q_X$. 
May 21 
comment 
How long can a cycle of antichains in a finite partial order be?
The bijection $L_X$ is wellknown and appears in many papers. (You seem to be using $L_X$ in two different ways.) It has been called the FonderFlaass action, the Panyushev action, and rowmotion. One reference is arxiv.org/pdf/1108.1172v3.pdf. However, I don't know of any work directly relevant to the question here. 
May 17 
comment 
Uniform generation of Symmetric Plane Partitons
From what set of symmetric plane partitions do you want to choose a random one? 
May 13 
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I need to refind a reference on multigraded Hilbert series
The proof of Theorem 11.1 in Atiyah and Macdonald, Introduction to Commutative Algebra, carries over to the multigraded case. It is also an easy consequence of the Hilbert syzygy theorem. 
May 12 
comment 
Interesting applications of the Pigeonhole Principle
I have reposted this at math.mit.edu/~rstan/pigeon.pdf. 
May 6 
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Combinatorial polynomials from general diagram fillings?
For socalled %avoiding diagrams, see the paper by Reiner and Shimozono at citeseerx.ist.psu.edu/viewdoc/…. 
Apr 30 
comment 
Number of matrices with given Smith normal form
Yinghui Wang has just written a paper on this topic. Soon it should be posted on the arXiv. 
Apr 29 
awarded  Good Answer 
Apr 28 
answered  Residue for the generating function of the Euler totient function 
Apr 27 
comment 
Generalization of Schur polynomials
There are some listed on page 405 of Enumerative Combinatorics, vol.2, that don't seem to be on your list. 
Apr 25 
comment 
A digraph related to permutations
One reason that the number of Hamiltonian cycles divided by $(n1)!^{(n1)!}$ factors quite a bit is the $S_n$symmetry. A similar situation is at OEIS oeis.org/A120061. See also Chapter 10, Exercise 7(d) of my book Algebraic Combinatorics. 
Apr 25 
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Is anything known about the eigenspectrum of the regular representation of the permutation group?
@Anirbit: Stembridge's Theorem 3.3 gives a combinatorial formula for the eigenvalues. The characters of $S_n$ are not involved. 
Apr 25 
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A question on (odd) perfect numbers
You left out the essential condition that $k$ is odd. 
Apr 25 
answered  Is anything known about the eigenspectrum of the regular representation of the permutation group? 
Apr 24 
comment 
A digraph related to permutations
Continuation of previous comment: $02167802121430577869247906928263206218129022043438732817393$ $78238299321477251081703396523732122511001097003078223354740$ $21115543570842542767571682953869289112205514506240000000000000000000$. This must have a large prime factor since Maple has been taking many hours trying to factor it. 
Apr 24 
comment 
A digraph related to permutations
For $n=7$ the number is $6!^{720}\cdot 176183413608273968258307195020261201995935498443817037880077$ $33015680734978977985994540951427241494888517320111715796104$ $01678883391824293568187890362411864333732042160720070054426$ $13973761520215432676342816642484823001649942753538184971817$ $00932653194251714270416078406552688838875907139437076885891$ $46604444975789292489211082309639859152636172062362845942377$ $16134182843447651691194901389583522511049879097164330759592530013$ (continued on next comment) 
Apr 23 
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A digraph related to permutations
For $n =6$ the number is $5!^{120}\cdot 2^{32}\cdot 3^{29} \cdot 5^{20}\cdot 7^9\cdot 23\cdot 37^3\cdot 53\cdot 79\cdot 83\cdot 311^9\cdot 1993\cdot 5569\cdot 57679$. 
Apr 23 
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A digraph related to permutations
For $n=5$ the number of Hamiltonian cycles is $4!^{24}\cdot 2^{11}\cdot 3^5\cdot 5^6\cdot 13^2\cdot 17\cdot 47$. 