Divisibility of sum of multinomials The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Integer-valued factorial ratiosPerron number distributionDesign constraint systems over the realsOn “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)Montgomery's conjecture and lower bound on certain Fourier transform.Distribution of the evaluation (at a non-trivial root of -1) of polynomials with small coefficientsCombinatorialProbabilistic Proof of Stirling's ApproximationGeneralizing Kasteleyn's formula even more?Derive a theoretical bound about coding with a partial eavesdropperTartaglia distribution
Divisibility of sum of multinomials
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Integer-valued factorial ratiosPerron number distributionDesign constraint systems over the realsOn “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)Montgomery's conjecture and lower bound on certain Fourier transform.Distribution of the evaluation (at a non-trivial root of -1) of polynomials with small coefficientsCombinatorialProbabilistic Proof of Stirling's ApproximationGeneralizing Kasteleyn's formula even more?Derive a theoretical bound about coding with a partial eavesdropperTartaglia distribution
$begingroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_k_1+cdots+k_n=mbinommk_1,dots,k_n^t$$
where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.
QUESTION. Is it always true that $n$ divides $S(n,m,t)$?
Observe that $S(n,m,1)=n^m$.
nt.number-theory co.combinatorics soft-question
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
$endgroup$
add a comment |
$begingroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_k_1+cdots+k_n=mbinommk_1,dots,k_n^t$$
where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.
QUESTION. Is it always true that $n$ divides $S(n,m,t)$?
Observe that $S(n,m,1)=n^m$.
nt.number-theory co.combinatorics soft-question
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
$endgroup$
add a comment |
$begingroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_k_1+cdots+k_n=mbinommk_1,dots,k_n^t$$
where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.
QUESTION. Is it always true that $n$ divides $S(n,m,t)$?
Observe that $S(n,m,1)=n^m$.
nt.number-theory co.combinatorics soft-question
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
$endgroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_k_1+cdots+k_n=mbinommk_1,dots,k_n^t$$
where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.
QUESTION. Is it always true that $n$ divides $S(n,m,t)$?
Observe that $S(n,m,1)=n^m$.
nt.number-theory co.combinatorics soft-question
nt.number-theory co.combinatorics soft-question
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
edited Apr 8 at 5:17
T. Amdeberhan
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
asked Apr 8 at 5:01
T. AmdeberhanT. Amdeberhan
18.3k230132
18.3k230132
add a comment |
add a comment |
1 Answer
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$begingroup$
We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of $1,ldots,m$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
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1 Answer
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1 Answer
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$begingroup$
We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of $1,ldots,m$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
$endgroup$
add a comment |
$begingroup$
We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of $1,ldots,m$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
$endgroup$
add a comment |
$begingroup$
We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of $1,ldots,m$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
$endgroup$
We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of $1,ldots,m$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
answered Apr 8 at 6:12
Fedor PetrovFedor Petrov
52.1k6122239
52.1k6122239
add a comment |
add a comment |
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