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Discrete mathematics and mathematical cybernetics
On recursion relations in the problem of enumeration of posets
V. I. Rodionov Udmurt State University, 1, Universitetskaya str., Izhevsk, 426034, Russia
Abstract:
In two previous works of the author, published in this journal, a series of formulas are obtained related to the themes of enumeration of partial orders (finite topologies). In the first work, a formula is proved that reduces the calculation of the number
$T_0(n)$ of all partial orders defined on an
$n$-set to the calculation of the numbers
$W(p_1,\ldots,p_k)$ of partial orders of a special form. In the second paper, a partially convolute formula is obtained for the number
$T_0(n)$. Relations of a recurrent nature are obtained that relate the individual values
$W(p_1,\ldots,p_k).$ Explicit formulas are presented for calculating the individual values
$W(p_1,\ldots,p_k). $ In this paper, we obtain new recurrence relations that relate the separate numbers
$W(p_1,\ldots,p_k)$ between themselves. The obtained equations are enough to calculate without the computer the numbers
$T_0(n)$ for all
$n<9.$ To calculate the number
$T_0(9)$ of these relations not enough (the number of required numbers
$W(p_1,\ldots,p_k)$ is
$128$, and the rank of the system matrix is
$123$; there are not enough five equations generating the desired rank). We admit the presence of some general regularity generating new formulas.
Keywords:
graph enumeration, poset, finite topology.
UDC:
519.175
MSC: 05C30 Received December 2, 2019, published
February 25, 2020
DOI:
10.33048/semi.2020.17.014