A000569 -id:A000569 - cp's OEIS frontend

A011968 Apply (1+Shift) to Bell numbers.

1, 2, 3, 7, 20, 67, 255, 1080, 5017, 25287, 137122, 794545, 4892167, 31858034, 218543759, 1573857867, 11863100692, 93345011951, 764941675963, 6514819011216, 57556900440429, 526593974392123, 4981585554604074, 48658721593531669, 490110875149889635

Offset: 0

N. J. A. Sloane

nonn

Number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n. The maximum number of singletons is therefore 3. Alternatively, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 3 (or n+2 if n+2 < 3). For example, a(3)=7 counts the following set partitions of [5]: {1245, 3}, {12, 3, 45}, {124, 3, 5}, {15, 24, 3}, {125, 3, 4}, {14, 25, 3}, {12, 3, 4, 5}. - Olivier Gérard, Oct 29 2007

Let V(N)={v(1),v(2),...,v(N)} denote an ordered set of increasing positive integers containing a pair of adjacent elements that differ by at least 2, that is, v(i),v(i+1) with v(i+1)-v(i) > 1. Then for n > 0, a(n) is the number of partitions of V(n+1) into blocks of nonconsecutive integers. - Augustine O. Munagi, Jul 17 2008

			a(3)=7 because the set {1,3,4,5} has 7 different partitions into blocks of nonconsecutive integers: 14/35, 135/4, 1/35/4, 13/4/5, 14/3/5, 15/3/4, 1/3/4/5.

Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011

Chai Wah Wu, Table of n, a(n) for n = 0..500 n = 0..200 from Vincenzo Librandi.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A000296, A011969, A011970.
A diagonal of A011971 and A106436. - N. J. A. Sloane, Jul 31 2012
Cf. A000569, A240936, A321750.

with(combinat): seq(`if`(n>0,bell(n)+bell(n-1),1),n=0..21); # Augustine O. Munagi, Jul 17 2008

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011968_list, blist, b = [1,2], [1], 1
for _ in range(10**2):
    blist = list(accumulate([b]+blist))
    A011968_list.append(b+blist[-1])
    b = blist[-1] # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

For n >= 1, a(n+1) = exp(-1)*Sum_{k>=0} ((k+1)/k!)*k^n. - Benoit Cloitre, Mar 09 2008
For n >= 1, a(n) = Bell(n) + Bell(n-1). - Augustine O. Munagi, Jul 17 2008
G.f.: G(0) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
G.f.: 1 + x*E(0) where E(k) = 1 + 1/(1-x*k-x)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 20 2013
G.f.: 1 + Sum_{k>=0} ( 1+1/(1-x-x*k) )*x^(k+1)/Product_{i=0..k} (1-x*i). - Sergei N. Gladkovskii, Jan 20 2013
a(n) ~ Bell(n) * (1 + LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

A321979 Number of e-positive simple labeled graphs on n vertices.

Original entry on oeis.org

1, 1, 2, 8, 60, 899

Offset: 0

Gus Wiseman, Nov 23 2018

A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895). A graph is e-positive if, in the expansion of its chromatic symmetric function in terms of elementary symmetric functions, all coefficients are nonnegative.

			The 4 non-e-positive simple labeled graphs on 4 vertices are:
  {{1,2},{1,3},{1,4}}
  {{1,2},{2,3},{2,4}}
  {{1,3},{2,3},{3,4}}
  {{1,4},{2,4},{3,4}}

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, The a(4) = 60 e-positive simple labeled graphs.

Cf. A000569, A006125, A229048, A240936, A277203, A321895, A321911, A321918, A321914, A321931, A321980, A321981, A321982.

A339839 Number of factorizations of n into distinct primes or semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 0, 2, 4, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 2, 4, 1, 1, 0, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 0, 1, 2, 2, 2, 1, 4, 1, 2, 4

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

A semiprime (A001358) is a product of any two prime numbers.

			The a(n) factorizations for n = 6, 16, 30, 60, 180, 210, 240, 420:
  6    5*6    4*15    4*5*9    6*35     4*6*10    2*6*35
  2*3  2*15   6*10    2*6*15   10*21    2*4*5*6   3*4*35
       3*10   2*5*6   2*9*10   14*15    2*3*4*10  4*5*21
       2*3*5  3*4*5   3*4*15   5*6*7              4*7*15
              2*3*10  3*6*10   2*3*35             5*6*14
                      2*3*5*6  2*5*21             6*7*10
                               2*7*15             2*10*21
                               3*5*14             2*14*15
                               3*7*10             2*5*6*7
                               2*3*5*7            3*10*14
                                                  3*4*5*7
                                                  2*3*5*14
                                                  2*3*7*10

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

A008966 allows only primes.
A320732 is the non-strict version.
A339742 does not allow squares of primes.
A339840 lists the positions of zeros.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A013929 cannot be factored into distinct primes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A322353 into distinct semiprimes.
- A339839 [this sequence] into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A000569 counts graphical partitions (A320922).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000070, A001222, A028260, A320893, A338898, A339617, A339661, A339740, A339741, A339846.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]],{n,100}]

A339839(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA339839(n/d, d))); (s)); \\ Antti Karttunen, Feb 10 2023

a(n) = Sum_{d|n squarefree} A322353(n/d).

Data section extended up to a(105) by Antti Karttunen, Feb 10 2023

A322109 Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 49, 50, 54, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 125, 126, 128, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 180, 189, 192, 196, 198, 200, 210

Offset: 1

Gus Wiseman, Nov 26 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also Heinz numbers of partitions whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

			Each term paired with its Heinz partition and a realizing set multipartition with no singletons:
   1:      (): {}
   4:    (11): {{1,2}}
   8:   (111): {{1,2,3}}
   9:    (22): {{1,2},{1,2}}
  12:   (211): {{1,2},{1,3}}
  16:  (1111): {{1,2,3,4}}
  18:   (221): {{1,2},{1,2,3}}
  24:  (2111): {{1,2},{1,3,4}}
  25:    (33): {{1,2},{1,2},{1,2}}
  27:   (222): {{1,2,3},{1,2,3}}
  30:   (321): {{1,2},{1,2},{1,3}}
  32: (11111): {{1,2,3,4,5}}
  36:  (2211): {{1,2},{1,2,3,4}}
  40:  (3111): {{1,2},{1,3},{1,4}}

These partitions are counted by A110618.
The even-weight version is A320924.
The conjugate case of equality is A340387.
The conjugate version is A344291.
The opposite conjugate version is A344296.
The opposite version is A344414.
The case of equality is A344415.
The opposite even-weight version is A344416.
A000070 counts non-multigraphical partitions.
A025065 counts palindromic partitions.
A035363 counts partitions into even parts.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A334201 adds up all prime indices except the greatest.
Cf. A000041, A000569, A265640, A283877, A306005, A318361, A320922, A320923, A320925.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]]
Select[Range[100],Length[sqnopfacs[Times@@Prime/@nrmptn[#]]]>0&]

A061395(a(n)) <= A056239(a(n))/2.

A029895 Number of partitions of floor(n^2/2) with at most n parts and maximal height n.

Original entry on oeis.org

1, 1, 2, 3, 8, 20, 58, 169, 526, 1667, 5448, 18084, 61108, 208960, 723354, 2527074, 8908546, 31630390, 113093022, 406680465, 1470597342, 5342750699, 19499227828, 71442850111, 262754984020, 969548468960, 3589093760726, 13323571588607, 49596793134484

Offset: 0

torsten.sillke(AT)lhsystems.com

nonn

This is the maximum value for the distribution of partitions of (0 .. n^2) that fit in an n X n box; assuming the peak of a normal distribution 1/sqrt(variance*2*Pi) approximates to these partitions and using A068606 suggests C(2n,n)*sqrt(6/(Pi*n^2*(2n+1))) could be an approximation [within 0.3% for a(100)=88064925963069745337300842293630181021718294488842002448]; using Stirling's approximation gives the simpler (sqrt(3)/Pi)*4^n/n^2 [about 0.6% away for a(100)] though experimentation suggests that something like (sqrt(3)/Pi)*4^n/(n^2+3n/5+1/5) is closer [about 0.0001% away for a(100)]. - Henry Bottomley, Mar 13 2002

Bisection of A277218 with even indexes. - Vladimir Reshetnikov, Oct 09 2016

			a(4)=8 because the partitions of Floor[4^2 /2] that fit inside a 4 X 4 box are {4, 4}, {4, 3, 1}, {4, 2, 2}, {4, 2, 1, 1}, {3, 3, 2}, {3, 3, 1, 1}, {3, 2, 2, 1}, {2, 2, 2, 2}.

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Vladimir Reshetnikov, Table of n, a(n) for n = 0..190
Eric W. Weisstein, q-Binomial Coefficient
Wikipedia, q-binomial

Cf. A000569, A004250, A004251, A029889, A047993, diagonal of A067059, A068607, A277218.

Table[Coefficient[Expand[FunctionExpand[QBinomial[2 n, n, q]]], q, Floor[n^2/2]], {n, 0, 30}] (* Vladimir Reshetnikov, Oct 09 2016 *)

{a(n)=if(n==0,1,polcoeff(prod(i=1,n,(1-q^(n+i))/(1-q^i)),n^2\2,q))} \\ Paul D. Hanna, Feb 15 2007

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser. Table[T[Floor[n^2/2], n, n], {n, 0, 36}] with T[ ] defined as in A047993. a(n)=A067059(n, n).

a(n) equals the central coefficient of q in the central q-binomial coefficients for n>0: a(n) = [q^([n^2/2])] Product_{i=1..n} (1-q^(n+i))/(1-q^i), with a(0)=1. - Paul D. Hanna, Feb 15 2007

More terms and comments from Wouter Meeussen, Aug 14 2001
Edited by Henry Bottomley, Feb 17 2002
a(27)-a(28) from Alois P. Heinz, Oct 31 2018

A320925 Heinz numbers of connected multigraphical partitions.

Original entry on oeis.org

4, 9, 12, 25, 27, 30, 36, 40, 49, 63, 70, 75, 81, 84, 90, 100, 108, 112, 120, 121, 147, 154, 165, 169, 175, 189, 196, 198, 210, 220, 225, 243, 250, 252, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 343, 351, 352, 360, 361, 363, 364, 385, 390, 400, 441

Offset: 1

Gus Wiseman, Oct 24 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is connected and multigraphical if it comprises the multiset of vertex-degrees of some connected multigraph.

			The sequence of all connected multigraphical partitions begins: (11), (22), (211), (33), (222), (321), (2211), (3111), (44), (422), (431), (332), (2222), (4211), (3221), (3311), (22211), (41111), (32111).

Cf. A000070, A000569, A007717, A056239, A112798, A147878, A209816, A300061, A320459, A320911, A320921, A320923, A320924.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Select[Range[1000],Select[prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]],Length[csm[#]]==1&]!={}&]

A322064 Number of ways to choose a stable partition of a simple connected graph with n vertices.

Original entry on oeis.org

1, 1, 1, 7, 141, 6533, 631875, 123430027, 48659732725, 39107797223409, 64702785181953175, 221636039917857648631, 1575528053913118966200441, 23249384407499950496231003021, 711653666389829384034090082068939, 45128328085994437067694854477617868995

Offset: 0

Gus Wiseman, Nov 25 2018

nonn

A stable partition of a graph is a set partition of the vertices where no non-singleton edge has both ends in the same block.

			The a(3) = 7 stable partitions. The simple connected graph is on top, and below is a list of all its stable partitions.
  {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}
  --------       --------       --------       --------
  {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}
  {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..75

Row sums of A322278.

Cf. A000110, A000569, A001187, A006125, A048143, A229048, A240936, A245883, A277203, A321911, A321979, A322063, A322065.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Sum[Length[Select[Subsets[Complement[Subsets[Range[n],{2}],Union@@Subsets/@stn]],And[Union@@#==Range[n],Length[csm[#]]==1]&]],{stn,sps[Range[n]]}],{n,5}]

\\ See A322278 for M.
seq(n)={concat([1], (M(n)*vectorv(n,i,1))~)} \\ Andrew Howroyd, Dec 01 2018

Terms a(7) and beyond from Andrew Howroyd, Dec 01 2018

A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 7, 1, 1, 25, 37, 5, 1, 1, 75, 207, 85, 21, 1, 1, 231, 1347, 525, 591, 7, 1, 1, 763, 10125, 21385, 23551, 3535, 113, 1, 1, 2619, 86173, 180201, 1216701, 31647, 30997, 9, 1, 1, 9495, 819133, 12066705, 77636583, 66620631, 11485825, 286929, 955, 1

Offset: 1

Gus Wiseman, Dec 17 2018

			Triangle begins:
  1
  1       1
  1       3       1
  1       9       7       1
  1      25      37       5       1
  1      75     207      85      21       1
  1     231    1347     525     591       7       1
  1     763   10125   21385   23551    3535     113       1
  1    2619   86173  180201 1216701   31647   30997       9       1

Andrew Howroyd, Table of n, a(n) for n = 1..300

Row sums are A322555.

Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.

Table[If[k==0,1,Sum[Binomial[n,sup]*SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[sup],{2}]}],Sequence@@Table[{x[i],0,k},{i,sup}]],{sup,n}]],{n,8},{k,0,n-1}]

T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - Andrew Howroyd, Dec 26 2020

A321980 Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 2, 2, 0, 0, 5, 3, 7, 1, 0, 0, 0, 6, 10, 4, 6, 2, 0, 4, 0, 0, 0, 0, 7, 5, 13, 17, 6, 0, 11, 4, 1, 0, 0, 0, 0, 0, 0, 8, 6, 16, 12, 0, 22, 16, 8, 12, 20, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 9, 7, 19, 27, 0, 31, 10, 9, 21, 0, 58, 16, 12, 9, 0

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).
All terms are nonnegative [Stanley].

			Triangle begins:
  1
  2  0
  3  1  0
  4  2  2  0  0
  5  3  7  1  0  0  0
  6 10  4  6  2  0  4  0  0  0  0
  7  5 13 17  6  0 11  4  1  0  0  0  0  0  0
  8  6 16 12  0 22 16  8 12 20  2  0  0  6  0  0  0  0  0  0  0  0
For example, row 6 gives: X_P6 = 6e(6) + 10e(42) + 4e(51) + 6e(33) + 2e(222) + 4e(321).

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Row sums are A000079.

Cf. A000569, A001187, A001349, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321981, A321982.

A321982 Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

Original entry on oeis.org

2, 0, 12, 2, 0, 0, 0, 54, 26, 16, 0, 2, 0, 0, 0, 0, 0, 0, 216, 120, 168, 84, 0, 24, 40, 32, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 810, 648, 822, 56, 240, 870, 280, 282, 120, 24, 0, 266, 232, 0, 48, 0, 54, 0, 48, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).
The n-ladder has 2*n vertices and looks like:
  o-o-o-   -o
  | | | ... |
  o-o-o-   -o
Conjecture: All terms are nonnegative (verified up to the 5-ladder).

			Triangle begins:
    2   0
   12   2   0   0   0
   54  26  16   0   2   0   0   0   0   0   0
  216 120 168  84   0  24  40  32   0   0   2   0   0   [+9 more zeros]
For example, row 3 gives: X_L3 = 54e(6) + 26e(42) + 16e(51) + 2e(222).

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Row sums are A109808.

Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321980, A321981.

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A011968 Apply (1+Shift) to Bell numbers.

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A321979 Number of e-positive simple labeled graphs on n vertices.

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A339839 Number of factorizations of n into distinct primes or semiprimes.

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A322109 Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

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A029895 Number of partitions of floor(n^2/2) with at most n parts and maximal height n.

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A320925 Heinz numbers of connected multigraphical partitions.

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A322064 Number of ways to choose a stable partition of a simple connected graph with n vertices.

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