You are on the bottom floor (floor 0, lets call it) of an apartment building with no basement. There are n elevators, which we index 1,2,…,n. Assume the elevators are on floors f1, f2, … , where fk > 0 is the floor elevator k is currently on, 1 <= k <= n. Assume you only like one of the elevators, elevator e.
The way the elevator logic works is this: When you press the elevator button, one closest to you ( = one on floor number min(fk, k>0)) is told to go to 0. If there is a tie then, of those on the same lowest floor, the elevator with the smallest index is told to go to 0.
Move: If you press the button and some other elevator than elevator e arrives, you can tell it to go to any floor you wish.
Taboo: You can press the elevator button if and only if no elevator is moving down.
Goal: You want to use elevator e (for some fixed e=1, 2, …, n).
Problem: Is there a finite sequence of moves that allows you to ride in elevator e?
My plan is to post the answer sometime later, but have fun with it!
be a simple, connected graph with vertices 
and 
adjacency matrix 
. We start with the geometric series identity
is the 
denote the orthonormal matrix of normalized eigenvectors, so that
, 
.![\frac{1}{I-tD_\Gamma}= P\cdot [\sum_{\ell=0}^\infty t^\ell A^\ell]\cdot P^{-1}.](https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7BI-tD_%5CGamma%7D%3D+P%5Ccdot+%5B%5Csum_%7B%5Cell%3D0%7D%5E%5Cinfty+t%5E%5Cell+A%5E%5Cell%5D%5Ccdot+P%5E%7B-1%7D.&bg=ffffff&fg=323232&s=0&c=20201002)
(i.e., 
in 
, we get,![\sum_{j=0}^{n-1} \lambda_j^{-1}H(f)(\lambda_j^{-1}) = {\frac{1}{\pi}}\sum_{\ell=0}^\infty tr(A^\ell) [M(f)(\ell+1)+(-1)^\ell M(f^*)(\ell+1)],](https://s0.wp.com/latex.php?latex=%5Csum_%7Bj%3D0%7D%5E%7Bn-1%7D+%5Clambda_j%5E%7B-1%7DH%28f%29%28%5Clambda_j%5E%7B-1%7D%29+%3D+%7B%5Cfrac%7B1%7D%7B%5Cpi%7D%7D%5Csum_%7B%5Cell%3D0%7D%5E%5Cinfty+tr%28A%5E%5Cell%29+%5BM%28f%29%28%5Cell%2B1%29%2B%28-1%29%5E%5Cell+M%28f%5E%2A%29%28%5Cell%2B1%29%5D%2C&bg=ffffff&fg=323232&s=0&c=20201002)
denotes the Hilbert transform
and 
is the Mellin transform
denotes the negation, 
. Of course, if 
is even then 
, for all 
. 
can be expressed in terms of the number of walks on the graph: If 
denotes the total number of walks of length 
, where 
is a quotient graph obtained from some subgroup 
. The examples are for graphs having a small number of vertices (no more than 12). For the most part, we also focused on regular graphs with small degree (no more than 5). They were all computed using 
.
is a harmonic morphism. Let 
be adjacent vertices of 
. Then either (a) 
and 
“collapses” the edge (vertical) 
or (b) 
and the vertices 
and 
are adjacent in 
. In the particular case of this post (ie, the case of 
the white vertex is not adjacent to the blue or red vertex, none of the harmonic colored graphs below can have a white vertex adjacent to a blue or red vertex.
as the domain in this post. However, before we get to examples (obtained by using 
be a harmonic morphism from a graph 
vertices to the path graph having 
vertices. Let 
be the coloring map (identified with an n-tuple whose coordinates are in 
). Associated to f is a partition ![\Pi_f=[n_0,\dots,n_{k-1}]](https://s0.wp.com/latex.php?latex=%5CPi_f%3D%5Bn_0%2C%5Cdots%2Cn_%7Bk-1%7D%5D&bg=ffffff&fg=323232&s=0&c=20201002)
of n (here ![[...]](https://s0.wp.com/latex.php?latex=%5B...%5D&bg=ffffff&fg=323232&s=0&c=20201002)
is a multi-set, so repetition is allowed but the ordering is unimportant): 
, where 
is the number of times j occurs in f. We call this the partition invariant of the harmonic morphism. 
, 
, with associated
whose corresponding partitions agree, 
then we say 
and 
examples!
, so we start with 
. We indicate a harmonic morphism by a vertex coloring. An example of a harmonic morphism can be described in the plot below as follows: 
sends the red vertices in 
, plus that induced by 
and all of its cyclic permutations (4+6=10). This set of 6 permutations is closed under the automorphism of 
and all of its cyclic permutations (4+7=11). This set of 7 permutations is not closed under the automorphism of 
and all 7 of its cyclic permutations (total = 7+11 = 18).
and all of its cyclic permutations (4+8=12). This set of 8 permutations is not closed under the automorphism of 
and all of its cyclic permutations (12+8=20). In addition, there is 
and all of its cyclic permutations (20+8 = 28). The latter set of 8 cyclic permutations of 
is closed under the transposition (0,3)(1,2) (total = 28).
and all of its cyclic permutations (4+9=13). This set of 9 permutations is not closed under the automorphism of 
and all 9 of its cyclic permutations (9+13 = 22). This set of 9 permutations is not closed under the automorphism of 
and all 9 of its cyclic permutations (9+22 = 31). This set of 9 permutations is not closed under the automorphism of 
and all 9 of its cyclic permutations (total = 9+31 = 40). 
is the graph whose SageMath command is 
then, instead of showing a graph, I’ll list the edges (of course, the vertices are 0,1,…,11) and the SageMath command for it.
, where 
.
.
.
.
.
.
.
.
and 
are graphs then a map 
) is a morphism provided
and 
then 
is an edge in 
and 
, where 
then 
.
is the dual map 
and 
, then 
, an edge 
is called horizontal if 
and is called vertical if 
. We say that a graph morphism 
is a graph homomorphism if 
. Thus, a graph morphism is a homomorphism if it has no vertical edges.
denote the star subgraph centered at the vertex v. A graph morphism 
is called harmonic if for all vertices 
, the quantity
and mapping to the edge 
.
sends the red vertices in 
and a card game called mathematical blackjack (which has almost no relation with the usual 
, a Steiner system S(k,m,n) is an n-set X and a set S of m-subsets having the property that any k-subset of X is contained in exactly one m-set in S. For example, if 
, a Steiner system S(5,6,12) is a set of 6-sets, called hexads, with the property that any set of 5 elements of X is contained in (“can be completed to”) exactly one hexad.
of X sends S to another Steiner system 
of X. It is known that if S and S’ are any two Steiner systems of type (5,6,12) in X then there is a 
such that 
. In other words, a Steiner system of this type is unique up to relabelings. (This also implies that if one defines 
to be the stabilizer of a fixed Steiner system of type (5,6,12) in X then any two such stabilizer groups, for different Steiner systems in X, must be conjugate in 
. In particular, such a definition is well-defined up to isomorphism.)
denote the projective line over the finite field 
with 11 elements. Let 
denote the quadratic residues with 0, and let 
where 
and 
. Let 
is a Steiner system of type 
.
6
2 10
5 7 3
6 9 4 6
2 10 8 2 10
0 1
arrays may be regarded as the plane 
. The lines of this plane are described by one of the following patterns.
,
using a $4\times 6$ array, called the MOG, Conway [Co1] found and analogous description of 
array, called the MINIMOG. This section is devoted to the MINIMOG. The tetracode words are
in any order.
hexads with total 
and none with lower total.
is another hexad.
and none with higher total.
(for example: king, ace, 
, 
, …, 
, jack, where the king is 
(to be fair, this should be done randomly). Each of the 
Yet Another Mathblog 
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