... y-ordinate from the bottom edge of Q
k+1
to the top edge of Q
k
. For Q in Diagr am 3,
δ(Q) =
2 3 6 4 4 5 3 2
3 5 5 2 6 5 4 1
.
The map δ is a bijection from CCP to Y. As such, δ allows CCPs to be ... to a bipartition of N
2
. In this
section, we use their Pattern Algebra to obtain a q-analog of Kitaev’s [25] Theorem 30
and to deduce a better generating function for per...
... combinatorics 3 (1996), #R16 17
[GL1] R. L. Graham and B. D. Lubachevsky, Dense packings of equal disks in an equilateral
triangle: from2 2to3 4andbeyond,The Electronic Journ. of Combinatorics ... =2,3,4,5,6 ,and7 (n = 5, 10, 18,
27, 39, and 52) and those packings (shown in Figs 2.9, 2.2, 2.5, and 2.8) are, indeed, optimal,
except the case of n =10: proved[GMPW]forn = 5, 10, 18 and...
... generating function for the
number of (132)-avoiding permutations that have a given number of (123) patterns,
and show how to extend this to permutations that have exactly one (132) pattern.
We ... numerator and denominator.
First, from (3) we have that
P (q, z, t)=
1
1−zP(q,zt, tq)
, (5)
the electronic journal of combinatorics 6 (1999), #R38 6
Finally, to find the denominator...
... Combinatorics on words. — Reading, Addison-Wesley, (Encyclop edia
of Math. and its Appl., 17).
[Mac13] MacMahon (Percy Alexander). — The indices of permutations and the derivation therefrom
of ... de toutes les multipermutations sign´ees, en r´eduisant le nombre des
variables. Ceci fait l’objet du paragraphe suivant.
9. Fonction g´en´eratrice de toutes les multipermutations sign´ee...
... Systems, and
by a grant from the Israel Science Foundation. We thank Uri Gavish for introducing us
to the combinatorial literature, and Brendan McKay and Herbert Wilf for their interest
and support. ... indebted to Gregory Berkolaiko for his idea concerning the proof of
(7) and (8), and to Akalu Tefera for his kind help in obtaining a computer-aided proof of
(2).
Reference...
... Mansour and A. Vainshtein, Layered restrictions and Chebychev polynomials
(2000), Annals of Combinatorics 5 (2001) 451–458.
[MV4] T. Mansour and A. Vainshtein, Restricted permutations and Chebyshev ... West, and G. Xin, Wilf–equivalence for singelton classes, in Proc.
13th Conf. in Formal Power Series and Algebraic Combinatorics, Tempe 2001.
[Bo1] M. B´ona, The permutation class...
... 1324-avoiding
permutations for length up to 20
Definition 5. Two permutations π and σ are said to be in the same strong class if the
left -to right minima of π are the same as those of σ andtheyoccurinthesameposition;
and ... n,then
k + 1 has to be the second entry of σ ;and2 )ifk + 1 occurs to the right of a
1
=1,then
k + 1 has to be the next -to- last entry of σ. Hence, 13...
... a
1
<a
2
and a
3
<a
4
(note that s
3
and s
4
belong to
the same horizontal edge); since there are at least two long horizontal edges and s
2
and
s
3
belong respectively to the first and last ... µ
−1
ν
p
and n = kp. We need to calculate how
many operators reduce to each one of the four scenarios in (4.9). Clearly λ =1⇔ µ = ν,
so a non-trivial operator will reduce t...
... d(Π,σ(Π,k,n)) = δ(Π,k,n). To find δ(Π), we will need to
find δ(Π,k,n), hence maximal Π-containing permutations σ(Π,k,n) are of interest to us,
especially, their asymptotic shape as n → and k →∞.
Example ... packing density problems from permutations to
patterns with repeated letters and generalized patterns. We are able to find the
packing density for some classes of patterns...