Divisibility of sum of multinomialsInteger-valued factorial ratiosPerron number distributionDesign constraint...
Divisibility of sum of multinomials
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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
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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
$endgroup$
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
$endgroup$
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
edited 1 hour ago
T. Amdeberhan
asked 1 hour ago
T. AmdeberhanT. Amdeberhan
18.3k229132
18.3k229132
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add a comment |
1 Answer
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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 $ncdot M$.
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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 $ncdot M$.
$endgroup$
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 $ncdot M$.
$endgroup$
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 $ncdot M$.
$endgroup$
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
51.9k6122239
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