All Questions
Tagged with integer-sequences partitions
16 questions
6
votes
0
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171
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An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
2
votes
1
answer
217
views
Number of distinct higher dimensional integer partitions
By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by ...
- 281
3
votes
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120
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Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,948
1
vote
0
answers
100
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Conjecture on numbers $k$ having only one partition into parts with same binary weight as a binary weight of $k$
Let $\operatorname{tr}(n)$ be A007814, number of trailing zeros in the binary representation of $n$.
Also, let $\operatorname{ntr}(n)$ be A086784, number of non-trailing zeros in the binary ...
- 4,948
3
votes
1
answer
92
views
Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $a(n,m)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)=m$. ...
- 4,948
8
votes
0
answers
318
views
Why are these Littlewood-Richardson coefficients congruent to 1 mod 8?
Let $n\in{\mathbb N}$ and write $n=q_1+q_2+\dots+q_t$, where $q_1>q_2>\dots>q_t$ are powers of $2$. Let $\lambda_n$ be the partition with Frobenius symbol $(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{t-1}...
- 1,090
2
votes
1
answer
133
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Conjectural congruences for numbers related to Littlewood-Richardson coefficients
For $n \geq 0$, let $a_n$ be the square of the Euclidean length of the vector of Littlewood-Richardson coefficients of $\sum_{\lambda \vdash n} s_\lambda^2$, where $s_\lambda$ are the symmetric Schur ...
- 19.7k
2
votes
0
answers
137
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Writing integers as sequences of products by 2 and integer divisions by 3
For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$.
For instance:
$$
100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
- 1,444
1
vote
1
answer
139
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(Translation request) Hypotheses of the Blom-Fredberg bounds on denumerants?
I don't know Swedish and I'm not finding the article "G. Blom and C. E. Froberg, On money changing" translated into English... so I tried to read the original (Swedish) with the help of ...
- 13
3
votes
1
answer
91
views
Source coding lexicographic index of finite alphabet sequence with weight (partitions)
My goal is to determine the lexicographic index of an $M$-ary $n$-sequence $\mathbf{x}$ on the subset with an $M$-weight sum constraint:
$$S = \{ \mathbf{x} \in \{0, \ldots, M-1\}^n: \sum_{j=1}^n x_j =...
- 131
2
votes
0
answers
154
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Equi-distribution of the parity of partitions
The integer partition function $p(n)$ has a generating function given by
$$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$
with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem ...
- 43.2k
3
votes
1
answer
329
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Counting Bipartitions
Numerical evidence suggests that $p_2(n) \geq n p(n)$ for large $n$.
Here $p(n)$ is the number of partitions of $n$, and $p_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of ...
- 133
0
votes
0
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224
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Classic question on integer partitions (with distinct summands)
I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem:
Denote $R(N,L)$ ...
- 1
4
votes
2
answers
240
views
Databases for sequences indexed by partitions
Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such ...
- 16.9k
13
votes
0
answers
718
views
Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable $\mod 4$?
Is A001935 Number of partitions with no even part repeated efficiently computable $\mod 4$?
I am interested because of this relation with sum of divisors of $8n+1$.
$\sigma(8n+1) \equiv A001935(n) \...
- 25.4k
11
votes
1
answer
864
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Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated )
Up to $10^6$:
$\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$
A001935 Number of partitions with no even part repeated
Is this true in general?
It would mean relation between restricted partitions ...
- 25.4k