Divisibility of sum of multinomialsInteger-valued factorial ratiosPerron number distributionDesign constraint...
Divisibility of sum of multinomials
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$begingroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
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 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
$endgroup$
add a comment |
$begingroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
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 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
$endgroup$
add a comment |
$begingroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
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 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
$endgroup$
Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
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 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
asked 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
edited 1 hour ago
T. Amdeberhan
asked 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
asked 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
asked 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
18.3k229132
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 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
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1 Answer
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1 Answer
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active
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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 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
$endgroup$
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 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
$endgroup$
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 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
$endgroup$
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 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
answered 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
answered 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
answered 41 mins ago
Fedor PetrovFedor Petrov
51.9k6122239
51.9k6122239
add a comment |
add a comment |
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