A note on divisors of multinomial coefficients

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F15%3A00316071" target="_blank" >RIV/68407700:21230/15:00316071 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s00013-015-0770-5" target="_blank" >https://doi.org/10.1007/s00013-015-0770-5</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00013-015-0770-5" target="_blank" >10.1007/s00013-015-0770-5</a>

Result language
angličtina
Original language name
A note on divisors of multinomial coefficients
Original language description
We introduce a simple equivalence relation on ordered rooted tree graphs. As a consequence we show that $$frac{(n_0 + n_1 + n_2 + dots + n_m - 1)!}{n_0 ! n_1 ! n_2 ! ldots n_m!}$$ is divisible by ({n_0 + 1}) , where ({n, n_0, n_1, n_2 ldots , n_m}) are nonnegative integers such that ({n - 1 = n_1 + 2n_2 + cdots + mn_m, n_0 = n - (n_1 + n_2 + cdots + n_m)}) . There is at least one ({a in {n_0 + 1, n_i mid i > 0}}) such that ({a}) is an odd positive integer, and for every divisor ({d > 1}) of every ({i + 1}) where ({n_i > 0}) and ({i > 0}) , there is at least one ({b in U_i = {n_0 + 1, n_j, n_i - 1 mid j > 0 {rm and} j not = i}}) which is not divisible by ({d}) . In particular, it follows that ({C_j equiv 0 pmod {j + 2}}) , where ({j > 2}) is an odd integer such that ({j - 1}) is not divisible by 3 and ({C_j}) denotes the ({j}) th Catalan number.
Czech name
—
Czech description
—

Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Publication year
2015
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Similar results(10)