%I A000262 M2950 N1190
%S A000262 1,1,3,13,73,501,4051,37633,394353,4596553,58941091,824073141,
%T A000262 12470162233,202976401213,3535017524403,65573803186921,1290434218669921,
%U A000262 26846616451246353,588633468315403843,13564373693588558173
%N A000262 Number of "sets of lists": number of partitions of {1,..,n} into any
number of lists, where a list means an ordered subset.
%C A000262 Determinant of n X n matrix M=[m(i,j)] where m(i,i)=i, m(i,j)=1 if i>
j, m(i,j)=i-j if j>i. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan
19 2003
%C A000262 a(n) = Sum_{k=0..n} |A008275(n,k)| * A000110(k). - Vladeta Jovovic (vladeta(AT)eunet.rs),
Feb 01 2003
%C A000262 a(n) = (n-1)!*LaguerreL(n-1,1,-1). - Vladeta Jovovic (vladeta(AT)eunet.rs),
May 10 2003
%C A000262 With p(n) = the number of integer partitions of n, d(i) = the number
of different parts of the i-th partition of n, m(i,j) = multiplicity
of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum
over i and prod_{j=1}^{d(i)} = product over j one has: a(n) = sum_{i=1}^{p(n)}
n!/(prod_{j=1}^{d(i)} m(i,j)!) - Thomas Wieder (wieder.thomas(AT)t-online.de),
May 18 2005
%C A000262 Consider all n! permutations of the integer sequence [n] = 1,2,3,...,
n. The i-th permutation, i=1,2,...,n!, consists of Z(i) permutation
cycles. Such a cycle has the length lc(i,j), j=1,...,Z(i). For a
given permutation we form the product of all its cycle lengths prod_{j=1}^{Z(i)}
lc(i,j). Furthermore, we sum up all such products for all permutations
of [n] which gives sum_{i=1 .. n!} prod_{j=1}^{Z(i)} lc(i,j) = A000262(n).
For n=4 we have sum_{i=1 .. n!} prod_{j=1}^{Z(i)} lc(i,j) = 1*1*1*1
+ 2*1*1 + 3*1 + 2*1*1 + 3*1 + 2*1 + 3*1 + 4 + 3*1 + 4 + 2*2 + 2*1*1
+ 3*1 + 4 + 3*1 + 2*1*1 + 2*2 + 4 + 2*2 + 4 + 3*1 + 2*1*1 + 3*1 +
4 = 73 = A000262(4). - Thomas Wieder (thomas.wieder(AT)t-online.de),
Oct 06 2006
%C A000262 For a finite set S of size n, a chain gang G of S is a partially ordered
set (S,<=) consisting only of chains. The number of chain gangs of
S is a(n). For example, with S={a, b}and n=2, there are a(2)=3 chain
gangs of S, namely, {(a,a),(b,b)}, {(a,a),(a,b),(b,b)} and {(a,a),
(b,a),(b,b)}. - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Feb 22 2007
%C A000262 (-1)*A000262 with the first term set to 1 and A084358 form a reciprocal
pair under the list partition transform and associated operations
described in A133314. Cf. A133289. - Tom Copeland (tcjpn(AT)msn.com),
Oct 21 2007
%C A000262 Consider the distribution of n unlabeled elements "1" onto n levels where
empty levels are allowed. In addition, the empty levels are labeled.
Their names are 0_1, 0_2, 0_3, etc. This sequence gives the total
number of such distributions. If the empty levels are unlabeled ("0"),
then the answer is A001700. Let the colon ":" separate two levels.
Then, for example, for n=3 we have a(3)=13 arrangements: 111:0_1:0_2,
0_1:111:0_2, 0_1:0_2:111, 111:0_2:0_1, 0_2:111:0_1, 0_2:0_1:111,
11:1:0, 11:0:1, 0:11:1, 1:11:0, 1:0:11, 0:1:11, 1:1:1. - Thomas Wieder
(thomas.wieder(AT)t-online.de), May 25 2008
%C A000262 Row sums of exponential Riordan array [1,x/(1-x)]. - Paul Barry (pbarry(AT)wit.ie),
Jul 24 2008
%C A000262 a(n) is the number of partitions of [n] (A000110) into lists of noncrossing
sets. For example, a(3)=3 counts 12, 1-2, 2-1 and a(4) = 73 counts
the 75 partitions of [n] into lists of sets (A000670) except for
13-24, 24-13 which fail to be noncrossing. - David Callan (callan(AT)stat.wisc.edu),
Jul 25 2008
%C A000262 a(i-j)/(i-j)! gives the value of the non-null element (i, j) of the lower
triangular matrix exp(S)/exp(1), where S is the lower triangular
matrix - of whatever dimension - having all its (non-null) elements
equal to one. [From Giuliano Ca rele (giulianoca rele(AT)tin.it),
Aug 11 2008, Sep 07 2008]
%C A000262 a(n) is also the number of nilpotent partial one-one bijections (of an
n-element set). Equivalently, it is the number of nilpotents in the
symmetric inverse semigroup (monoid) [From A. Umar (aumarh(AT)squ.edu.om),
Sep 14 2008]
%C A000262 A000262 is the exp transform of the factorial numbers A000142. [From
Thomas Wieder (thomas.wieder(AT)t-online.de), Sep 10 2008]
%C A000262 Contribution from David Angell (angell(AT)maths.unsw.edu.au), Dec 18
2008: (Start)
%C A000262 If n is a positive integer then the infinite continued fraction
%C A000262 (1+n)/(1+(2+n)/(2+(3+n)/(3+...)))
%C A000262 converges to the rational number A052852(n)/A000262(n). (End)
%D A000262 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000262 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000262 D. E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
%D A000262 T. S. Motzkin, Sorting numbers for cylinders and other classification
numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971,
pp. 167-176; the sequence called {!}^{n+}.
%D A000262 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 194.
%D A000262 Mark A. Shattuck and Carl G. Wagner, Parity Theorems for Statistics on
Lattice Paths and Laguerre Configurations, Journal of Integer Sequences,
Vol. 8 (2005), Article 05.5.1.
%D A000262 Laradji, A. and Umar, A. On the number of nilpotents in the partial symmetric
semigroup. Comm. Algebra 32 (2004), 3017-3023. [From A. Umar (aumarh(AT)squ.edu.om),
Sep 14 2008]
%H A000262 T. D. Noe, <a href="b000262.txt">Table of n, a(n) for n = 0..100</a>
%H A000262 P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon,
<a href="http://arXiv.org/abs/quant-ph/0501155">Boson normal ordering
via substitutions and Sheffer-type polynomials</a>
%H A000262 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/
abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized
Bell Numbers</a>
%H A000262 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/
abs/quant-ph/0402027">The general boson normal ordering problem.</
a>
%H A000262 David Callan, <a href="http://front.math.ucdavis.edu/0711.4841">Sets,
Lists and Noncrossing Partitions </a>.
%H A000262 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000262 F. Hivert, J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/
abs/math.CO/0605262">Commutative combinatorial Hopf algebras</a>
%H A000262 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=40">
Encyclopedia of Combinatorial Structures 40</a>
%H A000262 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
On generalizations of Stirling number triangles</a>, J. Integer Seqs.,
Vol. 3 (2000), #00.2.4.
%H A000262 T. S. Motzkin, <a href="a000262_1.png">Sorting numbers for cylinders
and other classification numbers</a> (part 1)
%H A000262 T. S. Motzkin, <a href="a000262_2.png">Sorting numbers for cylinders
and other classification numbers</a> (part 2)
%H A000262 T. S. Motzkin, <a href="a000262_3.png">Sorting numbers for cylinders
and other classification numbers</a> (part 3)
%H A000262 T. S. Motzkin, <a href="a000262_4.png">Sorting numbers for cylinders
and other classification numbers</a> (part 4)
%H A000262 T. S. Motzkin, <a href="a000262_5.png">Sorting numbers for cylinders
and other classification numbers</a> (part 5)
%H A000262 T. S. Motzkin, <a href="a000262_6.png">Sorting numbers for cylinders
and other classification numbers</a> (part 6)
%H A000262 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a
href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type
relations via substitution and the moment problem</a> [J. Phys. A
37 (2004), 3475-3487]
%H A000262 N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/
0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004),
83-89.
%H A000262 Thomas Wieder, <a href="a000262.txt">Further comments on this sequence</
a>
%H A000262 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000262 <a href="Sindx_La.html#Laguerre">Index entries for sequences related
to Laguerre polynomials</a>
%H A000262 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A000262 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
Publications/books.html">Analytic Combinatorics</a>, 2009; see page
125
%F A000262 a(n)=(2n-1)a(n-1) - (n-1)(n-2)a(n-2). E.g.f.: exp( x/(1-x) ).
%F A000262 Representation as n-th moment of a positive function on positive half-axis,
in Maple notation: a(n)=int(x^n*exp(-x-1)*BesselI(1, 2*x^(1/2))/x^(1/
2), x =0..infinity), n=1, 2... - Karol A. Penson, penson(AT)lptl.jussieu.fr
Dec 4 2003.
%F A000262 a(n) = Sum_{k=0..n} A001263(n, k)*k!. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Dec 10 2003
%F A000262 a(n) = n! Sum[Binomial[n-1, j]/(j+1)!, {j, 0, n-1}] for n=1, 2, 3, ...
- Herbert S. Wilf (wilf(AT)math.upenn.edu), Jun 14 2005
%F A000262 Asymptotic expansion for large n: a(n)->(0.4289*n^(-1/4)+0.3574*n^(-3/
4)-0.2531*n^(-5/4)+O(n^(-7/4)))*(n^n)*exp(-n+2*sqrt(n)) .- Karol
A. Penson (penson(AT)lptl.jussieu.fr), Aug 28 2002
%F A000262 a(n) = exp(-1)*Sum_{m>=0} [m]^n/m!, where [m]^n = m*(m+1)*...*(m+n-1)
is the rising factorial. - Vladeta Jovovic (vladeta(AT)eunet.rs),
Sep 20 2006
%F A000262 Recurrence: D(n,k) = D(n-1,k-1) + (n-1+k) * D(n-1,k) n >= k >= 0; D(n,
0)=0. From this, D(n,1) = n! and D(n,n)=1; a(n) = Sum[from i=0 to
n] D(n,i). - Stephen Dalton (StephenMDalton(AT)gmail.com), Jan 05
2007
%F A000262 Proof: Notice two distinct subsets of the lists for [n]: 1) n is in its
own list, then there are D(n-1,k-1); 2) n is in a list with other
numbers. Denoting the separation of lists by |, it is not hard to
see n has (n-1+k) possible positions, so (n-1+k) * D(n-1,k) - Stephen
Dalton (StephenMDalton(AT)gmail.com), Jan 05 2007
%F A000262 Define f_1(x),f_2(x),... such that f_1(x)=e^x, f_{n+1}(x)=diff(x^2*f_n(x),
x), for n=2,3,.... Then a(n-1)=e^{-1}*f_n(1). - Milan R. Janjic (agnus(AT)blic.net),
May 30 2008
%F A000262 a(n) = (n-1)! sum_{k=1}^{n} (a(n-k) k!)/((n-k)! (k-1)!) where a(0)=1.
[From Thomas Wieder (thomas.wieder(AT)t-online.de), Sep 10 2008]
%e A000262 a(2) = 3: (12), (21), (1)(2). a(4) = 73: (1234) (24 ways), (123)(4) (6*4
ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1
way).
%e A000262 a(2) = 3 from {1}{2}; {1,2}; {2,1}
%p A000262 a := proc(n) option remember: if n=0 then RETURN(1) fi: if n=1 then RETURN(1)
fi: (2*n-1)*a(n-1) - (n-1)*(n-2)*a(n-2) end:for n from 0 to 20 do
printf(`%d,`,a(n)) od:
%p A000262 spec := [S, {S = Set(Prod(Z,Sequence(Z)))}, labeled]; [seq(combstruct[count](spec,
size=n), n=0..40)];
%p A000262 with(combinat):seq(sum(abs(stirling1(n, k))*bell(k), k=0..n), n=0..18);
- Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 26 2006
%p A000262 B:=[S,{S = Set(Sequence(Z,1 <= card),card <=13)},labelled]: seq(combstruct[count](B,
size=n), n=0..19);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Mar 21 2009]
%t A000262 Range[0, 19]! CoefficientList[ Series[E^(x/(1 - x)), {x, 0, 19}], x]
(from Robert G. Wilson v Apr 04 2005)
%t A000262 a[n_]:=If[n<0, 0, n!*SeriesCoefficient[Series[Exp[x/(1-x)], {x, 0, n}],
n]] (Somos)
%t A000262 a[n_]:=If[n=0, 1, n! Sum[Binomial[n-1, j]/(j+1)!, {j, 0, n-1}]] (Wilf)
%o A000262 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(x/(1-x)+x*O(x^n)),n))
%o A000262 (PARI) a(n)=if(n<0,0,n!*polcoeff(prod(k=1,n, eta(x^k+x*O(x^n))^(-moebius(k)/
k)),n)) /* Michael Somos Feb 10 2005 */
%Y A000262 a(n), n >= 1, is sum of n-th row of A008297 (unsigned Lah-triangle) -
comment from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
%Y A000262 A002868 = maximal element of n-th row of A008297. Cf. A066668.
%Y A000262 Cf. A001263.
%Y A000262 A111596 (unsigned row sums of triangle).
%Y A000262 Cf. A001700.
%Y A000262 A052852 [From David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008]
%Y A000262 Sequence in context: A067764 A063512 A132846 this_sequence A059294 A124468
A128196
%Y A000262 Adjacent sequences: A000259 A000260 A000261 this_sequence A000263 A000264
A000265
%K A000262 nonn,easy,core,nice
%O A000262 0,3
%A A000262 N. J. A. Sloane (njas(AT)research.att.com).
%E A000262 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 06 2000
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