A097237 -id:A097237 - cp's OEIS frontend

A097236 Number of hierarchical orderings ("societies") with at least 2 elements ("individuals") on each level for n labeled elements.

0, 1, 1, 10, 31, 271, 1534, 14393, 117653, 1253524, 13140557, 160679069, 2026451948, 28278471729, 413532314433, 6516434058758, 107958571213579, 1899723866781859, 35092386753388698, 682552407940860353, 13893916425860413469, 296049402365644855888

Offset: 1

Thomas Wieder, Aug 02 2004

nonn

			a(4) = 10. Let : denote the partition of n labeled individuals 1,2,3,4 into x=2 sets (i.e. "societies"). E.g., in 12:34 one has a single society with members 1 and 2 and a further single society with members 3 and 4. Let | denote a higher level (within a single society), e.g., in 1|2 the individual 2 is one level up with respect to individual 1. The order of individuals on a level is insignificant, e.g. 12|34 is equivalent to 21|43.
For n = 4 and x = 2 one has 1234; 12:34; 13:24; 14:23; 12|34; 31|42; 43|21; 24|13; 21|34; 43|12; which gives 10 different hierarchical societies with at least 2 labeled individuals per level.

Alois P. Heinz, Table of n, a(n) for n = 1..175
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Cf. A075729, A097237, A226572.

with(combstruct); SetSeqSetxL:=[T,{T=Set(S), S=Sequence(U,card>=1), U=Set(Z,card >= 2)},labeled];
# where x is an integer 1, 2, 3,... ; x=2 gives 2 individuals per level.
seq (count (SetSeqSetxL,size=j), j=1..20);

terms = 22;
CoefficientList[ Exp[-(-Exp[z]+1+z)/(2-Exp[z]+z)] + O[z]^(terms+1), z] *  Range[0, terms]! // Rest (* Jean-François Alcover, Aug 06 2018 *)

E.g.f.: exp(-(-exp(z)+1+z)/(2-exp(z)+z)).

a(n) ~ exp(1/(2*(c-2)) + 1/(2*(c-1)^2) + 2*sqrt(n/((c-2)*(c-1))) - n - 1) * n^(n-1/4) / (sqrt(2) * (c-1)^(1/4) * (c-2)^(n+1/4)), where c = -LambertW(-1, -exp(-2)) = A226572 = 3.14619322062... . - Vaclav Kotesovec, Sep 08 2014

A101052 Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed.

Original entry on oeis.org

1, 1, 3, 13, 51, 181, 603, 1933, 6051, 18661, 57003, 173053, 523251, 1577941, 4750203, 14283373, 42915651, 128878021, 386896203, 1161212893, 3484687251, 10456158901, 31372671003, 94126401613, 282395982051, 847221500581

Offset: 0

Thomas Wieder, Nov 28 2004

nonn

The (labeled) case for k <= 2 is given by A000225. The unlabeled analog for k <= 2 is given by A028310 (A000027). The unlabeled analog for k <= 3 is given by A000124.

Alice and Bob went out for dinner; Alice paid 10 euros for the taxi, Bob paid 20 euros for the dinner; if they have to equally divide the expenses Alice will have to give 5 euros to Bob. With two people, Alice and Bob, there are three possible cases: Alice has to give money to Bob; Bob has to give money to Alice; they paid the same amount, so no debtors nor creditors. With three people, there are 13 cases; with four people, there are 51 cases, and so on. - Alessandro Gentilini (alessandro.gentilini(AT)gmail.com), Aug 10 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hongyi Cao, Gang Xu, Renshu Gu, Jinlan Xu, Xiaoyu Zhang, Timon Rabczuk, Yuzhe Luo, and Xifeng Gao, Robust and Feature-Preserving Offset Meshing, arXiv:2412.15564 [cs.GR], 2024. See p. 14.
Samuele Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938, 2013
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000670, A000225, A000124, A028310, A097237.

Cf. A000918.

A101052 := n -> 3^n+2-2*2^n; [ seq(3^n+2-2*2^n,n=0..30) ];

a = Exp[x] - 1;CoefficientList[Series[1+a+a^2+a^3,{x,0,20}],x]*Table[n!,{n,0,20}]
LinearRecurrence[{6,-11,6},{1,1,3},30] (* Harvey P. Dale, Mar 13 2013 *)

E.g.f. = 2*exp(z) - 2*exp(z)^2 + exp(z)^3;
o.g.f. = -(-1+3*z-6*z^2)/(11*z^2+1-6*z-6*z^3).
a(n) = 3^n + 2 - 2*2^n; recurrence: a(n+3) - 6*a(n+2) + 11*a(n+1) - 6*a(n), a(0) = 1, a(1) = 1, a(2) = 3.
G.f.: Sum_{n>=0} a(n)*log(1+x)^n/n! = (1-x^4)/(1-x). - Paul D. Hanna, Feb 18 2012
Binomial transform of A000918 in which the first term is changed from -1 to 1 as: (1, 0, 2, 6, 14, 30, 62, ...). - Gary W. Adamson, Mar 23 2012

A097391 The number of hierarchies of n labeled elements with at least one subhierarchy composed of exactly 2 levels and no subhierarchy with more than 2 levels.

Original entry on oeis.org

0, 1, 3, 8, 17, 37, 71, 138, 252, 458

Offset: 1

Thomas Wieder, Aug 13 2004

Conjectured to be (A378854) /3. - Wouter Meeussen, Feb 10 2025

			Let : denote the separation between two subhierarchies, e.g. 2:3 are two subhierarchies where subhierarchy s=1 contains two elements and subhierarchy s=2 contains three elements. Let | denote the separation between two levels, e.g. 2|2|1 is a hierarchy composed of three levels with two elements on levels l=1 and l=2 and one element on level l=3. For n=5 one has a(5) = 17 hierarchies where at least one subhierarchy has exactly 2 levels (and no level l > 2 is allowed):
4|1; 1|4; 3|2; 2|3; 2|2:1; 2|1:2; 1|2:2; 2|1:1|1; 1|2:1|1; 2|1:1:1; 1|2:1:1; 1|1:1:1:1; 1|3:1; 3|1:1; 1|1:2:1; 1|1:1|1:1; 1|1:3.

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A034691, A097237, A097392, A000041, A378854.

A110045 Number of hierarchical orderings ("societies") of n unlabeled elements ("individuals") with at least two occupied levels.

Original entry on oeis.org

1, 0, 1, 3, 8, 18, 45, 102, 245, 565, 1324, 3049, 7066, 16199, 37187, 84887, 193532, 439600, 996818, 2253941, 5086980, 11454778, 25746467, 57756522, 129342179, 289153474, 645399011, 1438308839, 3200671082, 7112360474, 15783402471, 34980122720, 77428353682

Offset: 0

Thomas Wieder, Jul 09 2005

nonn

Unlabeled analog of A097237.

Primes in this sequence include: a(3) = 3, a(11) = 3049, a(19) = 2253941, a(22) = 25746467. Semiprimes in this sequence include: a(9) = 565 = 5 * 113, a(12) = 7066 = 2 * 3533, a(13) = 16199 = 97 * 167, a(14) = 37187 = 41 * 907, a(15) = 84887 = 11 * 7717, a(18) = 996818 = 2 * 498409, a(24) = 129342179 = 23 * 5623573, a(30) = 15783402471 = 3 * 5261134157. - Jonathan Vos Post, Jul 10 2005

			Let * denote an unlabeled element.
Let : denote a delimiter between two levels of a hierarchy.
Let | denote a delimiter between two subhierarchies.
a(4) = 8 because we have *:*:*:*, ***:*, **:*:*, *:*|*:*, *:***, **:**, *:**:*, *:*:**.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A075729, A034691, A097237.

SetSeqXSetU := [S, {S=Set(U), U=Sequence(V,card>=2),V=Set(Z,card>=1)},unlabeled]; seq(count(SetSeqXSetU,size=j),j=0..30); #where x is an integer 1, 2, 3,... # x=2 gives 2 levels per society.

nmax = 40; CoefficientList[Series[E^Sum[x^(2*k)/(k*(1 - x^k)*(1 - 2*x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 08 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^(2^(k-1)-1). - Ilya Gutkovskiy, Jun 07 2018

a(n) ~ 2^n * exp(sqrt(2*n) - 5/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} 1/(k*(2^k-1)*(2^k-2)) = 0.0927294481510243482503144824759369647388... - Vaclav Kotesovec, Jun 08 2018

A097392 The number of hierarchies of n labeled elements with at least one subhierarchy composed of exactly 3 levels and no subhierarchy with more than 3 levels.

Original entry on oeis.org

0, 0, 1, 4, 12, 32, 78, 183, 408, 886

Offset: 1

Thomas Wieder, Aug 13 2004

			Let : denote the separation between two subhierarchies, e.g. 2:3 are two subhierarchies where subhierarchy s=1 contains two elements and subhierarchy s=2 contains three elements. Let | denote the separation between two levels, e.g. 2|2|1 is a hierarchy composed of three levels with two elements on levels l=1 and l=2 and one element on level l=3. For n=5 one has a(5) = 12 hierarchies where at least one subhierarchy has exactly 3 levels (and no level l > 3 is allowed):
3|1|1; 1|3|1; 1|1|3; 2|2|1; 2|1|2; 1|2|2; 1|1|1:2; 1|1|1:1:1; 1|1|1:1|1; 2|1|1:1; 1|2|1:1; 1|1|2:1.

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A034691, A097237, A097391, A000041.

cp's OEIS Frontend

A097236 Number of hierarchical orderings ("societies") with at least 2 elements ("individuals") on each level for n labeled elements.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A101052 Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A097391 The number of hierarchies of n labeled elements with at least one subhierarchy composed of exactly 2 levels and no subhierarchy with more than 2 levels.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A110045 Number of hierarchical orderings ("societies") of n unlabeled elements ("individuals") with at least two occupied levels.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A097392 The number of hierarchies of n labeled elements with at least one subhierarchy composed of exactly 3 levels and no subhierarchy with more than 3 levels.

Views

Author

Keywords

Examples

Links

Crossrefs