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Revision History for A157018

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A157018

Triangle T(n,k) read by rows: number of k-lists (ordered k-sets) of disjoint 2-subsets of an n-set, n>1, 0<k<=floor(n/2).
(history; published version)

#15 by N. J. A. Sloane at Sat Nov 05 08:13:03 EDT 2016

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proposed

approved

#14 by Michael De Vlieger at Fri Nov 04 18:52:18 EDT 2016

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editing

proposed

#13 by Michael De Vlieger at Fri Nov 04 18:52:15 EDT 2016

MATHEMATICA

Table[n!/(2^k (n - 2 k)!), {n, 2, 13}, {k, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 04 2016 *)

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proposed

editing

#12 by Stanislav Sykora at Fri Nov 04 04:26:45 EDT 2016

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editing

proposed

#11 by Stanislav Sykora at Fri Nov 04 04:24:09 EDT 2016

COMMENTS

T(n,k) is also the number of involutions (unary operators) on S_n, i.e., endomorfisms endomorphisms U with 2k non-invariant elements such that U^2 is the identity mapping. The extension to n=1 is a(1)=0. - Stanislav Sykora, Nov 03 2016

Discussion

Fri Nov 04

04:26

Stanislav Sykora: Yes, thanks. I have also added a link, and priviledged the term 'involution' to 'unary operator'. The letter is what the physicist in me instinctively prefers but it is a bit dangerous because it can be confused with 'binary operation' which is something different.

#10 by Stanislav Sykora at Fri Nov 04 04:21:45 EDT 2016

COMMENTS

T(n,k) is also the number of involutions (unary operators (involutions) on S_n, i.e., endomorfisms U with 2k non-invariant elements such that U^2 is the identity mapping. The extension to n=1 is a(1)=0. - Stanislav Sykora, Nov 03 2016

LINKS

Wikipedia, <a href="http://en.wikipedia.org/wiki/Involution_(mathematics)">Involution (mathematics)</a>.

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proposed

editing

#9 by Michel Marcus at Fri Nov 04 01:22:47 EDT 2016

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editing

proposed

#8 by Michel Marcus at Fri Nov 04 01:22:24 EDT 2016

FORMULA

E.g.f.: y*x^2*exp(x)/(2-y*x^2). T(n,k) = Product_{m=1..floor(n/2)} binomial(n-2*m,2) = n!/(2^k*(n-2*k)!).

T(n,k) = Product_{m=1..floor(n/2)} binomial(n-2*m,2) = n!/(2^k*(n-2*k)!).

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proposed

editing

Discussion

Fri Nov 04

01:22

Michel Marcus: rather endomorphisms ?

#7 by Wesley Ivan Hurt at Thu Nov 03 11:47:46 EDT 2016

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editing

proposed

#6 by Wesley Ivan Hurt at Thu Nov 03 11:47:41 EDT 2016

EXAMPLE

For n = 4 we have 12 lists, : 6 1-lists: [{1,2}], [{1,3}], [{1,4}], [{2,3}], [{2,4}], [{3,4}] and 6 2-lists: [{1,2},{3,4}], [{3,4},{1,2}], [{1,3},{2,4}], [{2,4},{1,3}], [{1,4},{2,3}] and [{2,3},{1,4}].

CROSSREFS

Cf. A000262, A000680, A087214, A100861, A126725 (row sums), A087214, A129684, A000262, A100861, A000680.

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proposed

editing