A318284 -id:A318284 - cp's OEIS frontend

A322260 Numbers k such that the poset of multiset partitions of a multiset whose multiplicities are the prime indices of k is a lattice.

1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 32

Offset: 1

Gus Wiseman, Dec 05 2018

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

R. P Stanley, Enumerative Combinatorics Vol. 1, Sec. 3.3.

Wikipedia, Lattice (order)
Gus Wiseman, The first 9 lattices (n = 1, 2, ..., 9).
Gus Wiseman, The first 6 non-lattices (n = 10, 11, ..., 15).

Cf. A001055, A007716, A057716, A265947, A281113, A305936, A317144, A317534, A318284.

A321279 Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 4, 2, 2, 1, 2, 3, 4, 4, 2, 4, 3, 4, 4, 3, 4, 6, 4, 6, 2, 1, 4, 6, 4, 9, 6, 5, 3, 9, 2, 8, 4, 9, 8, 7, 4, 8, 4, 12, 6, 12, 5, 16, 8, 17, 5, 7, 2, 19, 6, 10, 10, 1, 6, 13, 2, 16, 7, 16, 6, 27, 4, 7, 16, 20, 8, 15, 4, 22

Offset: 1

Gus Wiseman, Nov 01 2018

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

			The sequence of antichains begins:
   2: {{1}}
   3: {{1,1}}
   3: {{1},{1}}
   4: {{1,2}}
   5: {{1,1,1}}
   5: {{1},{1},{1}}
   6: {{1,1,2}}
   7: {{1,1,1,1}}
   7: {{1,1},{1,1}}
   7: {{1},{1},{1},{1}}
   8: {{1,2,3}}
   9: {{1,1,2,2}}
  10: {{1,1,1,2}}
  10: {{1,1},{1,2}}
  11: {{1,1,1,1,1}}
  11: {{1},{1},{1},{1},{1}}
  12: {{1,1,2,3}}
  12: {{1,2},{1,3}}
  13: {{1,1,1,1,1,1}}
  13: {{1,1,1},{1,1,1}}
  13: {{1,1},{1,1},{1,1}}
  13: {{1},{1},{1},{1},{1},{1}}
  14: {{1,1,1,1,2}}
  14: {{1,2},{1,1,1}}
  15: {{1,1,1,2,2}}
  15: {{1,1},{1,2,2}}
  16: {{1,2,3,4}}

Cf. A001055, A007718, A030019, A181821, A293607, A303837, A304382, A305081, A305936, A318284, A321229, A321270, A321271, A321272.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[Times@@Prime/@nrmptn[n]],And[zensity[#]==-1,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]

A321745 Sum of coefficients of monomial symmetric functions in the homogeneous symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 5, 10, 16, 12, 7, 27, 11, 20, 32, 47, 15, 76, 22, 56, 65, 35, 30, 136, 79, 54, 263, 114, 42, 191, 56, 246, 113, 86, 160, 476, 77, 128, 199, 344

Offset: 1

Gus Wiseman, Nov 19 2018

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

Also the number of size-preserving permutations of multiset partitions of a multiset (such as row n of A305936) whose multiplicities are the prime indices of n.

			The sum of coefficients of h(211) = m(4) + 4m(22) + 3m(31) + 7m(211) + 12m(1111) is a(12) = 27.
The a(3) = 2 through a(9) = 16 size-preserving permutations of multiset partitions:
  {11}    {12}    {111}      {112}      {1111}        {123}      {1122}
  {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{111}      {1}{23}    {1}{122}
          {2}{1}  {1}{1}{1}  {2}{11}    {11}{11}      {2}{13}    {11}{22}
                             {1}{1}{2}  {1}{1}{11}    {3}{12}    {12}{12}
                             {1}{2}{1}  {1}{1}{1}{1}  {1}{2}{3}  {2}{112}
                             {2}{1}{1}                {1}{3}{2}  {22}{11}
                                                      {2}{1}{3}  {1}{1}{22}
                                                      {2}{3}{1}  {1}{2}{12}
                                                      {3}{1}{2}  {2}{1}{12}
                                                      {3}{2}{1}  {2}{2}{11}
                                                                 {1}{1}{2}{2}
                                                                 {1}{2}{1}{2}
                                                                 {1}{2}{2}{1}
                                                                 {2}{1}{1}{2}
                                                                 {2}{1}{2}{1}
                                                                 {2}{2}{1}{1}

Wikipedia, Symmetric polynomial

Row sums of A321744.

Cf. A005651, A007716, A008480, A056239, A124794, A124795, A181821, A255906, A296150, A318284, A319193, A319225, A319226, A321742-A321765.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,mps[nrmptn[n]]}],{n,30}]

A321913 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in h(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 6, 1, 1, 1, 1, 1, 1, 3, 2, 4, 6, 1, 2, 2, 3, 4, 1, 4, 3, 7, 12, 1, 6, 4, 12, 24, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 1, 2, 3, 5, 4, 7, 10, 1, 3, 5, 11, 8, 18, 30, 1, 3, 4, 8, 7, 13, 20, 1, 4, 7, 18, 13, 33, 60, 1, 5

Offset: 1

Gus Wiseman, Nov 22 2018

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

Also the coefficient of f(v) in e(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

			Tetrangle begins:
  (1):  1
.
  (2):   1  1
  (11):  1  2
.
  (3):    1  1  1
  (21):   1  2  3
  (111):  1  3  6
.
  (4):     1  1  1  1  1
  (22):    1  3  2  4  6
  (31):    1  2  2  3  4
  (211):   1  4  3  7 12
  (1111):  1  6  4 12 24
.
  (5):      1  1  1  1  1  1  1
  (41):     1  2  2  3  3  4  5
  (32):     1  2  3  5  4  7 10
  (221):    1  3  5 11  8 18 30
  (311):    1  3  4  8  7 13 20
  (2111):   1  4  7 18 13 33 60
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: h(32) = m(5) + 3m(32) + 2m(41) + 5m(221) + 4m(311) + 7m(2111) + 10m(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321744.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.

A321917 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in p(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 3, 6, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 2, 2, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 1, 2, 1, 0, 2, 0, 0, 1, 3, 4, 6, 6, 6, 0, 1, 5, 10, 30

Offset: 1

Gus Wiseman, Nov 22 2018

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

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   1
  (11):  1  2
.
  (3):    1
  (21):   1  1
  (111):  1  3  6
.
  (4):     1
  (22):    1  2
  (31):    1     1
  (211):   1  2  2  2
  (1111):  1  6  4 12 24
.
  (5):      1
  (41):     1  1
  (32):     1     1
  (221):    1  1  2  2
  (311):    1  2  1     2
  (2111):   1  3  4  6  6  6
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: p(32) = m(5) + m(32).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321750.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.

A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, 2, 3, 1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, -2, -1, -2, -1, 0, 6, 3, 8, 6, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 0, -6, -6, -5, -3, -3, -1, 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).

The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):  -1
  (11):  1  1
.
  (3):    1
  (21):  -1 -1
  (111):  2  3  1
.
  (4):    -1
  (22):    1  1
  (31):    1     1
  (211):  -2 -1 -2 -1
  (1111):  6  3  8  6  1
.
  (5):      1
  (41):    -1 -1
  (32):    -1    -1
  (221):    2  1  2  1
  (311):    2  2  1     1
  (2111):  -6 -6 -5 -3 -3 -1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: F(32) = -p(5) - p(32).

Wikipedia, Symmetric polynomial

Row sums are A178803. Up to sign, same as A321931. This is a regrouping of the triangle A321899.

Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 14, 15, 26, 22, 21, 29, 19, 30, 33, 31, 30, 66, 38, 42, 52, 56, 42, 47, 45, 57, 82, 77, 67, 77, 67, 101, 98, 135, 64, 137, 97, 176, 104, 109, 109, 118, 105, 231, 213, 97, 127, 181, 139, 297, 173, 385, 195, 269

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A multiset partition is crossing if it contains two blocks of the form {{...x...y...},{...z...t...}} where x < z < y < t or z < x < t < y.

			The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,4},{2,3}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{1,2},{4}}
  {{1},{2},{3},{4}}
Missing from this list is {{1,3},{2,4}}.

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324326.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

a(n) + A324326(n) = A318284(n).

A321915 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 1, -2, 1, 4, -2, -4, 4, -1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 4, -4, -7, 10, -3, -1, 1, 2, -3, 1, 5, -5, -5, 5, 5, -5, 1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, 5, -7, -11, 14, 10, -14, 3, 5, -9, -8, 10, 12

Offset: 1

Gus Wiseman, Nov 22 2018

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

Also the coefficient of e(v) in f(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

			Tetrangle begins:
  (1):  1
.
  (2):   2 -1
  (11): -1  1
.
  (3):    3 -3  1
  (21):  -3  5 -2
  (111):  1 -2  1
.
  (4):     4 -2 -4  4 -1
  (22):   -2  3  2 -4  1
  (31):   -4  2  7 -7  2
  (211):   4 -4 -7 10 -3
  (1111): -1  1  2 -3  1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):    -5  9  5 -7 -9  9 -2
  (32):    -5  5 11 11 -8 10 -2
  (221):    5 -7 11 14 10 14  3
  (311):    5 -9 -8 10 12 13  3
  (2111):  -5  9 10 14 13 17 -4
  (11111):  1 -2 -2  3  3 -4  1
For example, row 14 gives: m(32) = -5h(5) + 11h(32) + 5h(41) - 11h(221) - 8h(311) + 10h(2111) - 2h(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321748. Row sums are A155972.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.

A321916 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in h(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, -2, 1, 0, -1, 1, 0, 0, 1, -1, 1, 2, -3, 1, 0, 1, 0, -2, 1, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 1, -2, -2, 3, 3, -4, 1, 0, -1, 0, 1, 2, -3, 1, 0, 0, -1, 2, 1, -3, 1, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, -1, 1

Offset: 1

Gus Wiseman, Nov 22 2018

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

Also the coefficient of h(v) in e(u).

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -1  1
  (11):     1
.
  (3):    1 -2  1
  (21):     -1  1
  (111):        1
.
  (4):    -1  1  2 -3  1
  (22):       1    -2  1
  (31):          1 -2  1
  (211):           -1  1
  (1111):              1
.
  (5):      1 -2 -2  3  3 -4  1
  (41):       -1     1  2 -3  1
  (32):          -1  2  1 -3  1
  (221):             1    -2  1
  (311):                1 -2  1
  (2111):                 -1  1
  (11111):                    1
For example, row 14 gives: h(32) = -e(32) + 2e(221) + e(311) - 3e(2111) + e(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321749. Row sums are A134286.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A321919 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in p(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 2, -1, 0, 1, 3, -3, 1, 0, 2, -1, 0, 0, 1, 4, -2, -4, 4, -1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 1, 5, -5, -5, 5, 5, -5, 1, 0, 4, 0, -2, -4, 4, -1, 0, 0, 6, -6, -3, 5, -1, 0, 0, 0, 4, 0, -4, 1, 0, 0, 0, 0, 3, -3, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Nov 22 2018

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

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   2 -1
  (11):     1
.
  (3):    3 -3  1
  (21):      2 -1
  (111):        1
.
  (4):     4 -2 -4  4 -1
  (22):       4    -4  1
  (31):          3 -3  1
  (211):            2 -1
  (1111):              1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):        4    -2 -4  4 -1
  (32):           6 -6 -3  5 -1
  (221):             4    -4  1
  (311):                3 -3  1
  (2111):                  2 -1
  (11111):                    1
For example, row 14 gives: p(32) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321754.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

cp's OEIS Frontend

A322260 Numbers k such that the poset of multiset partitions of a multiset whose multiplicities are the prime indices of k is a lattice.

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A321279 Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

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A321745 Sum of coefficients of monomial symmetric functions in the homogeneous symmetric function of the integer partition with Heinz number n.

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A321913 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in h(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

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A321917 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in p(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

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A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

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A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A321915 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

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A321916 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in h(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

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A321919 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in p(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

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