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Results tagged with co.combinatorics
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user 1916
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
2
votes
Sets of equations
Here is a second attempt (see edit history for previous version).
For each $t\in\mathbb{N}$, let
$$P_{i,j,k,t}=\{1_{i,j,k,t},\ldots,n_{i,j,k,t},\ldots,\gamma(i,j,k)_{i,j,k,t}\}$$
(so that for each …
0
votes
Representing numbers in a non-integer base with few (but possibly negative) nonzero digits
Your proof looks right to me. As a smaller example, a number with a Zeckendorf representation with 7 terms is 609 = 377 + 144 + 55 + 21 + 8 + 3 + 1 (note that 609 = 610 - 1); I believe it is impossibl …
- 6,792
6
votes
1
answer
2k
views
Gamma function versions of combinatorial identites?
We can extend the binomial coefficient $\binom{n}{k}$ to $\mathbb{R}$ or $\mathbb{C}$ by defining $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$. Do any the standard binomial coefficient …
- 6,792
14
votes
3
answers
869
views
How can we realize different combinatorial objects as the dimension of a construction on vec...
Fix a vector space $V$ of dimension $n$ over some field $F$. Here are three commonly seen constructions:
its $k$th tensor power, $T^kV$, which has dimension $n^k$
its $k$th exterior power, $\Lambda^ …
- 6,792