A354847 - OEIS

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A354847

Number of binary relations on [n] that are idempotent and reduced.

1, 2, 6, 32, 318, 5552, 159126, 7137272, 484656318, 48628712192, 7076367228486, 1471524821492552, 432066672598422318, 177354805872559516112, 100928502119652298356726, 79062670900333522721886872, 84733519638342583432646258718, 123582326772837258238596562116512, 244150974458417420635453430918487846

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OFFSET

0,2

COMMENTS

The Boolean matrix representing a binary relation on [n] is row (column) reduced if no nonzero row (column) is the sum of other rows (columns). It is reduced if it is both row reduced and column reduced.

a(n) is the number of partial order relations on Y, where Y is some subset of [n].

LINKS

Table of n, a(n) for n=0..18.

R. J. Plemmons and M. T. West, On the semigroup of binary relations, Pacific Journal of Mathematics, vol 35, No. 3, 1970. Theorem 2.4

FORMULA

E.g.f.: A(x)*exp(x) where A(x) is the e.g.f. for A001035.

a(n) = Sum_{k=0..n} binomial(n,k)*A001035(n-k).

MATHEMATICA

nn = 18; A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt",

"Table"], {_, _}][[All, 2]]; A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, nn}];

Range[0, nn]! CoefficientList[Series[A[x] Exp[x], {x, 0, nn}], x]

CROSSREFS