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1.
A partition of an integer n is a representation n=a
1+a
2+⋅⋅⋅+a
k
, with integer parts 1≤a
1≤a
2≤…≤a
k
. For any fixed positive integer p, a p-succession in a partition is defined to be a pair of adjacent parts such that a
i+1−a
i
=p. We find generating functions for the number of partitions of n with no p-successions, as well as for the total number of such successions taken over all partitions of n. In the process, various interesting partition identities are derived. In addition, the Hardy-Ramanujan asymptotic formula
for the number of partitions is used to obtain an asymptotic estimate for the average number of p-successions in the partitions of n.
This material is based upon work supported by the National Research Foundation under grant number 2053740. 相似文献
2.
Shichao Chen 《The Ramanujan Journal》2009,18(1):103-112
Let Λ={λ
1≥⋅⋅⋅≥λ
s
≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ
i
nodes in the ith row. Let λ
j
′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ
i
+λ
j
′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2
t
-core partitions of n are obtained. A simple convolution identity for t-cores is also given.
相似文献
3.
N. S. Romanovskii 《Algebra and Logic》2011,49(6):539-550
Let ε = (ε
1, . . . , ε
m
) be a tuple consisting of zeros and ones. Suppose that a group G has a normal series of the form G = G
1 ≥ G
2 ≥ . . . ≥ G
m
≥ G
m+1 = 1, in which G
i > G
i+1 for ε
i = 1, G
i = G
i+1 for ε
i
= 0, and all factors G
i
/G
i+1 of the series are Abelian and are torsion free as right ℤ[G/G
i
]-modules. Such a series, if it exists, is defined by the group G and by the tuple ε uniquely. We call G with the specified series a rigid m-graded group with grading ε. In a free solvable group of derived length m, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid
m-graded groups. It is proved that the category of rigid m-graded groups contains coproducts, and we show how to construct a coproduct G◦H of two given rigid m-graded groups. Also it is stated that if G is a rigid m-graded group with grading (1, 1, . . . , 1), and F is a free solvable group of derived length m with basis {x
1, . . . , x
n
}, then G◦F is the coordinate group of an affine space G
n
in variables x
1, . . . , x
n
and this space is irreducible in the Zariski topology. 相似文献
4.
Raymond Viglione 《Acta Appl Math》2008,104(2):173-176
Let q be a prime power,
the field of q elements, and n≥1 a positive integer. The Wenger graph W
n
(q) is defined as follows: the vertex set of W
n
(q) is the union of two copies P and L of (n+1)-dimensional vector spaces over
, with two vertices (p
1,p
2,…,p
n+1)∈P and [l
1,l
2,…,l
n+1]∈L being adjacent if and only if l
i
+p
i
=p
1
l
i−1 for 2≤i≤n+1. Graphs W
n
(q) have several interesting properties. In particular, it is known that when connected, their diameter is at most 2n+2. In this note we prove that the diameter of connected Wenger graphs is 2n+2 under the assumption that 1≤n≤q−1. 相似文献
5.
A sequence (z
0,z
1,z
2,, ...,z
n, z
n+1) of points fromp=z
0 toq=z
n+1 in a metric spaceX is said to besequentially equidistant ifd(z
i−1,z
i)=d(z
i,z
i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR
m
,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the
positive results.
Supported in part by NSF Grant DMS-8701666. 相似文献
6.
Gustavo A. Fernández-Alcober 《Israel Journal of Mathematics》2007,162(1):75-79
Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ωi(G) ≤ p
i for odd p, and expΩi(G) ≤ 2
i+1 for p = 2; (ii) the index |G: G
p
i| coincides with the number of elements of G of order at most p
i.
Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University
of the Basque Country, grant UPV05/99. 相似文献
7.
D. V. Millionshchikov 《Proceedings of the Steklov Institute of Mathematics》2006,252(1):182-204
We study symplectic structures on filiform Lie algebras, which are niplotent Lie algebras with the maximal length of the descending
central sequence. Let g be a symplectic filiform Lie algebra and dim g = 2k ≥ 12. Then g is isomorphic to some ℕ-filtered deformation either of m0(2k) (defined by the structure relations [e
1, e
i
] = e
i+1, i = 2,…, 2k − 1) or of V
2k
, the quotient of the positive part of the Witt algebra W
+ by the ideal of elements of degree greater than 2k. We classify ℕ-filtered deformations of V
n
: [e
i
, e
j
] = (j − i)e
i+1 + Σ
l≥1
c
ij
l
e
i+j+l
. For dim g = n ≥ 16, the moduli space ℳn of these deformations is the weighted projective space
. For even n, the subspace of symplectic Lie algebras is determined by a single linear equation.
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 252, pp. 194–216. 相似文献
8.
A composition of a positive integer n is a finite sequence π1π2...π
m
of positive integers such that π1+...+π
m
= n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if π
i+1 ≥ π
i
+d (respectively, π
i
< π
i+1 < π
i
+ d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size
d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable generating function for the number of compositions of
n according to the number of parts, ascents of size d or more, ascents of size less than d, descents and levels. Also, we extend the results of Brennan and Knopfmacher to the case of ascents of size less than d. More precisely, we determine the mean, variance and limiting distribution of the number of ascents of size less than d in the set of compositions of n. 相似文献
9.
In this paper, we first consider a delay difference equation of neutral type of the form:
Δ(y
n
+ py
n−k
+ q
n
y
n−l
= 0 for n∈ℤ+(0) (1*)
and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form:
Δ1(u
n,m
+ pu
n−k,m
) + q
n,m
u
n−l,m
= a
2Δ2
2
u
n
+1,
m−1
for (n,m) ∈ℤ+ (0) ×Ω, (2*)
study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses.
Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000 相似文献
10.
11.
We show that every (possibly unbounded) convex polygon P in
\mathbbR2{\mathbb{R}^2} with m edges can be represented by inequalities p
1 ≥ 0, . . ., p
n
≥ 0, where the p
i
’s are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies s(m, k) £ n(m, k) £ (1+em) s(m, k){s(m, k) \leq n(m, k) \leq (1+\varepsilon_m) s(m, k)} with s(m,k) ≔ max {m/k, log2
m} and em ? 0{\varepsilon_m \rightarrow 0} as m ? ¥{m \rightarrow \infty}. This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p
1 > 0, . . ., p
n
> 0 is also given. For k ≤ m/log2
m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k. 相似文献
12.
In this paper we study three-color Ramsey numbers. Let K
i,j
denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G
1, G
2 and G
3, if r(G
1, G
2)≥s(G
3), then r(G
1, G
2, G
3)≥(r(G
1, G
2)−1)(χ(G
3)−1)+s(G
3), where s(G
3) is the chromatic surplus of G
3; (ii) (k+m−2)(n−1)+1≤r(K
1,k
, K
1,m
, K
n
)≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K
k,m
, K
k,m
, K
n
)≤c(n/logn), and r(C
2m
, C
2m
, K
n
)≤c(n/logn)
m/(m−1)
for sufficiently large n.
Received: July 25, 2000 Final version received: July 30, 2002
RID="*"
ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of
education ministry of China, and the foundations of Jiangsu Province
Acknowledgments. The authors are grateful to the referee for his valuable comments.
AMS 2000 MSC: 05C55 相似文献
13.
Jinkun LIN 《数学年刊B辑(英文版)》2006,27(3):311-328
This paper proves the existence of an order p element in the stable homotopy group of sphere spectrum of degree pnq pmq q - 4 and a nontrivial element in the stable homotopy group of Moore spectum of degree pnq pmq q - 3 which are represented by h0(hmbn-1-hnbm-1) and i*(h0hnhm) in the E2-terms of the Adams spectral sequence respectively, where p≥7 is a prime, n≥m 2≥4, q = 2(p - 1). 相似文献
14.
Let gzs(m, 2k) (gzs(m, 2k+1)) be the minimal integer such that for any coloring Δ of the integers from 1, . . . , gzs(m, 2k) by (the integers from 1 to gzs(m, 2k+1) by ) there exist integers
such that
1. there exists jx such that Δ(xi) ∈ for each i and ∑i=1m Δ(xi) = 0 mod m (or Δ(xi)=∞ for each i);
2. there exists jy such that Δ(yi) ∈ for each i and ∑i=1m Δ(yi) = 0 mod m (or Δ(yi)=∞ for each i); and
1. 2(xm−x1)≤ym−x1.
In this note we show gzs(m, 2)=5m−4 for m≥2, gzs(m, 3)=7m+−6 for m≥4, gzs(m, 4)=10m−9 for m≥3, and gzs(m, 5)=13m−2 for m≥2.
Supported by NSF grant DMS 0097317 相似文献
15.
In this paper we consider elliptic equations of order 2m in a bounded domainQ є R
n
with boundaryδQ and nonlocal conditions relating the traces of the solution and its derivatives on (n − 1)-dimensional smooth manifolds Γ
i
(∪
i
=∂δQ) to their values on some compact setF ⊂Q, whereF ∩δQ ≠ Φ. The Fredholm solvability of these problems in the weight spacesV
p, a
/l+2m
(Q) is proved for arbitrary 1<p <∞.
Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 882–898, June, 2000.
This research was supported by the Russian Foundation for Basic Research under grant No. 99-01-00028. 相似文献
16.
Gustavo A. Fernández-Alcober Jon González-Sánchez Andrei Jaikin-Zapirain 《Israel Journal of Mathematics》2008,166(1):393-412
Let G be a pro-p group and let k ≥ 1. If γ
k(p−1) (G) ≤ γ
r
for some r and s such that k(p − 1) < r + s(p − 1), we prove that the exponent of Ωi(G) is at most p
i+k−1 for all i.
Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds.
The first author is also supported by the University of the Basque Country, grant UPV05/99.
The second author is also supported by the Basque Government. 相似文献
17.
In this paper, we study the initial-boundary value problem of the porous medium equation u
t
= Δu
m
+ V(x)u
p
in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|)
σ
. Let ω
1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l
2 + (n − 2)l = ω
1. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u
0 unless u
0 = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u
0 ≥ 0. 相似文献
18.
Jin Kun Lin 《数学学报(英文版)》2008,24(3):471-490
This paper proves that, for any generator x∈ExtA^s,tq(Zp,Zp), if (1L ∧i)*Ф*(x)∈ExtA^s+1,tq+2q(H*L∧M, Zp) is a permanent cycle in the Adams spectral sequence (ASS), then b0x ∈ExtA^s+1,tq+q(Zp, Zp) also is a permenent cycle in the ASS. As an application, the paper obtains that h0hnhm∈ExtA^3,pnq+p^mq+q(Zp, Zp) is a permanent cycle in the ASS and it converges to elements of order p in the stable homotopy groups of spheres πp^nq+p^mq+q-3S, where p ≥5 is a prime, s ≤ 4, n ≥m+2≥4 and M is the Moore spectrum. 相似文献
19.
Brunetto Piochi 《Semigroup Forum》1995,50(1):301-306
LetS be a finitely generated semigroup. ThenS is finite if every finitely generated subgroup ofS is finite and, for some integerm≥1, for everym-tuplex
1,x
2,…x
m
of elements ofS there exist an integeri: 1≤i≤m and an integer ρ>1 such that:x
i
+1…x
m
(x
1
x
2…x
m
)ρ=x
i
+1…x
m
x
1…x
m
. The proof of the result is a direct generalisation of the original one by Green and Rees for the casem=1. 相似文献
20.
Domenico A. Catalano Marston D. E. Conder Shao Fei Du Young Soo Kwon Roman Nedela Steve Wilson 《Journal of Algebraic Combinatorics》2011,33(2):215-238
An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the
group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge
pairs). Such embeddings of the n-dimensional cubes Q
n
were classified for all odd n by Du, Kwak and Nedela in 2005, and in 2007, Jing Xu proved that for n=2m where m is odd, they are precisely the embeddings constructed by Kwon in 2004. Here, we give a classification of orientably-regular
embeddings of Q
n
for all n. In particular, we show that for all even n (=2m), these embeddings are in one-to-one correspondence with elements σ of order 1 or 2 in the symmetric group S
n
such that σ fixes n, preserves the set of all pairs B
i
={i,i+m} for 1≤i≤m, and induces the same permutation on this set as the permutation B
i
↦
B
f(i) for some additive bijection f:ℤ
m
→ℤ
m
. We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio
of reflexible to chiral embeddings tends to zero for large even n. 相似文献