A345317 Number of transitive but not symmetric relations on an n-set.

0, 0, 8, 156, 3942, 154100, 9414312, 878218390, 122207682476, 24890747805972, 7307450298831718, 3053521546328889460, 1797003559223742679800, 1476062693867018935173990, 1679239558149570227773844452, 2628225174143857306613215434524, 5626175867513779058706923151723150

Offset

0,3

Comments

The number of relations on n elements that are symmetric and transitive is the same as the number of equivalence relations on n+1 elements. Consequently, the number of relations that are transitive but not symmetric is obtained by subtracting the terms in the sequence of Bell numbers from that of the sequence of number of labeled transitive relations.

a(n) is even for all n.

Links

Table of n, a(n) for n=0..16.

Des MacHale and Peter MacHale, Relations on Sets, The Mathematical Gazette, Vol. 97, No. 539 (July 2013), pp. 224-233 (10 pages).

Firdous Ahmad Mala, Some New Integer Sequences of Transitive Relations, J. Appl. Math. Comp. (2023) Vol. 7, No. 1, 108-111.

Formula

a(n) = A006905(n) - A000110(n+1).

Example

For n=3, a(3) = A006905(3) - A000110(4) = 171 - 15 = 156.

Crossrefs

Cf. A006905, A000110.

Sequence in context: A089669 A288682 A268543 * A113668 A120348 A251586

Adjacent sequences: A345314 A345315 A345316 * A345318 A345319 A345320

Keyword

nonn

Author

Firdous Ahmad Mala, Oct 02 2021

Status

approved