A321738 -id:A321738 - cp's OEIS frontend

A321742 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

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

Offset: 1

Gus Wiseman, Nov 19 2018

Row n has length A000041(A056239(n)).

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

			Triangle begins:
   1
   1
   0   1
   1   2
   0   0   1
   0   1   3
   0   0   0   0   1
   1   3   6
   0   1   0   2   6
   0   0   0   1   4
   0   0   0   0   0   0   1
   0   2   1   5  12
   0   0   0   0   0   0   0   0   0   0   1
   0   0   0   0   0   1   5
   0   0   0   1   0   3  10
   1   6   4  12  24
   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1
   0   0   1   5   2  12  30
For example, row 12 gives: e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111).

Wikipedia, Symmetric polynomial

Row sums are A321743.

Cf. A008480, A049311, A056239, A116540, A124794, A124795, A300121, A319193, A321738, A321742-A321765, A321854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],And[And@@UnsameQ@@@#,Sort[Length/@#]==primeMS[k]]&]}],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}],{n,18}]

A321912 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in e(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 3, 1, 3, 6, 0, 0, 0, 0, 1, 0, 1, 0, 2, 6, 0, 0, 0, 1, 4, 0, 2, 1, 5, 12, 1, 6, 4, 12, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0, 0, 1, 0, 3, 10, 0, 0, 1, 5, 2, 12, 30, 0, 0, 0, 2, 1, 7, 20, 0, 1, 3, 12, 7, 27, 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 h(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):      1
  (11):  1  2
.
  (3):          1
  (21):      1  3
  (111):  1  3  6
.
  (4):                 1
  (22):       1     2  6
  (31):             1  4
  (211):      2  1  5 12
  (1111):  1  6  4 12 24
.
  (5):                        1
  (41):                    1  5
  (32):              1     3 10
  (221):          1  5  2 12 30
  (311):             2  1  7 20
  (2111):      1  3 12  7 27 60
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: e(32) = m(221) + 3m(2111) + 10m(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321742.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321913-A321935.

A321728 Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 50

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A000701 at a(11) = 28, A000701(11) = 27
A vertical section is a partial Young diagram with at most one square in each row.
Conjecture: a(n) is the number of non-half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(2) = 1 through a(9) = 14 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the non-half-loop-graphical partitions up to n = 9:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (31)  (32)  (33)   (43)   (44)    (54)
                  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (411)  (331)  (71)    (81)
                               (421)  (422)   (432)
                               (511)  (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (5111)  (621)
                                              (711)
                                              (4311)
                                              (5211)
                                              (6111)
For example, a complete list of all half/full-loop-graphs with degrees y = (4,3,1) is the following:
  {{1,1},{1,2},{1,3},{2,2}}
  {{1},{2},{1,1},{1,2},{2,3}}
  {{1},{2},{1,1},{1,3},{2,2}}
  {{1},{3},{1,1},{1,2},{2,2}}
None of these is a half-loop-graph, as they have full loops (x,x), so y is counted under a(8).

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

The complement is counted by A321729.
Cf. A000110, A000258, A000700, A000701, A008277, A046682, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339655.
A027187 counts partitions of even length, with Heinz numbers A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 counts graphical partitions of 2n into k parts.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]=={}&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is zero, where m is monomial and e is elementary symmetric functions.

a(n) = A000041(n) - A321729(n).

A321854 Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 19 2018

Row n has length A000041(A056239(n)).

A vertical section is a partial Young diagram with at most one square in each row.

			Triangle begins:
  1
  1
  0  1
  1  1
  0  0  1
  0  2  1
  0  0  0  0  1
  1  3  1
  0  2  0  4  1
  0  0  0  3  1
  0  0  0  0  0  0  1
  0  2  2  5  1
  0  0  0  0  0  0  0  0  0  0  1
  0  0  0  0  0  4  1
  0  0  0  6  0  6  1
  1  3  4  6  1
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
  0  0  4 10  4  8  1
The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):
  T(12,7) = 0:
.
  T(12,9) = 2:    1 2   1 2
                  1     2
                  2     1
.
  T(12,10) = 2:   1 2   1 2
                  2     1
                  2     1
.
  T(12,12) = 5:   1 2   1 2   1 2   1 2   1 2
                  3     2     3     1     3
                  3     3     2     3     1
.
  T(12,16) = 1:   1 2
                  3
                  4

Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Table[Length[Select[spsu[ptnverts[y],ptnpos[y]],Sort[Length/@#]==primeMS[k]&]],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]],{n,18}]

A321729 Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 28, 40, 51

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A046682 at a(11) = 28, A046682(11) = 29.
A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3
Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(1) = 1 through a(8) = 12 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the half-loop-graphical partitions up to n = 8:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (2221)     (2222)
                    (1111)  (2111)   (2211)    (3211)     (3221)
                            (11111)  (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, the half-loop-graphs
  {{1},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3}}
both have degrees y = (3,2,2), so y is counted under a(7).

The complement is counted by A321728.
Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A046682, A319056, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339656.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 is a triangle counting graphical partitions by length.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Length[Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]]>0&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is nonzero, where m is monomial symmetric functions and e is elementary symmetric functions.

a(n) = A000041(n) - A321728(n).

A321737 Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

Original entry on oeis.org

1, 1, 3, 9, 37, 152, 780, 3965, 23460, 141471, 944217, 6445643, 48075092, 364921557, 2974423953, 24847873439, 219611194148, 1987556951714, 18930298888792, 184244039718755, 1874490999743203, 19510832177784098, 210941659716920257, 2331530519337226199, 26692555830628617358

Offset: 0

Gus Wiseman, Nov 19 2018

nonn

A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
2 3
2
3

			The a(4) = 37 partitions into vertical sections of integer partitions of 4:
  1 2 3 4
.
  1 2 3   1 2 3   1 2 3   1 2 3
  4       3       2       1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3 4   2 3   3 2   1 3   1 2   3 1   2 1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1
.
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
  2   2   2   2   2   1   1   2   2   2   2   1   1   2   1
  3   3   2   3   2   2   2   1   1   3   2   1   2   1   1
  4   3   3   2   2   3   2   3   2   1   1   2   1   1   1

Cf. A000110, A000258, A008277, A046682, A122111, A318396, A321728, A321729, A321730, A321731, A321738, A321854.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[Sum[Length[spsu[ptnverts[y],ptnpos[y]]],{y,IntegerPartitions[n]}],{n,6}]

a(11)-a(24) from Ludovic Schwob, Aug 28 2023

A321730 Number of ways to partition the Young diagram of an integer partition of n into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 79, 303, 1294, 5934, 29385, 156232, 884893

Offset: 0

Gus Wiseman, Nov 18 2018

A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
2 3
2
3

			The a(5) = 23 partitions of Young diagrams of integer partitions of 5 into vertical sections of the same sizes as the parts of the original partition, shown as colorings by positive integers:
  1 2 3   1 2 3   1 2 3
  1       2       3
  1       2       3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1 2   1 3   1 3   2 1   3 1   3 1   2 3   3 2   2 3   3 2
  3     2     3     3     2     3     1     1     3     3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1     3     3     2     3     3     3     3     3
  3     1     4     3     2     4     3     4     4
  4     4     1     4     4     2     4     3     4
.
  1
  2
  3
  4
  5

Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A321728, A321729, A321731, A321737, A321738.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Sum[Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]],{y,IntegerPartitions[n]}],{n,5}]

A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

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

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 f(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

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

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321746.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321912-A321935.

A321731 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 5, 0, 0, 0, 1, 0, 10, 0, 3, 0, 0, 0, 9, 0, 0, 8, 0, 0, 12, 0, 1, 0, 0, 0, 34, 0, 0, 0, 10, 0, 0, 0, 0, 24, 0, 0, 14, 0, 0, 0, 0, 0, 68, 0, 4, 0, 0, 0, 78, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 86, 0, 0, 36, 0, 0, 0, 0, 22, 60, 0, 0

Offset: 1

Gus Wiseman, Nov 18 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3

			The a(30) = 12 partitions of the Young diagram of (321) into vertical sections of sizes (321), shown as colorings by positive integers:
  1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3
  1 2     1 3     2 1     3 1     1 2     1 3
  1       1       1       1       2       3
.
  1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3
  2 1     3 1     2 3     3 2     2 3     3 2
  2       3       2       2       3       3

Cf. A000110, A000258, A000700, A000701, A056239, A122111, A321649, A321728, A321729, A321730, A321737, A321738.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]]],{n,30}]

A321746 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 1, -2, 1, 1, 0, 3, -3, 1, -3, 1, 0, -4, 2, 4, -4, 1, 1, 0, 0, 2, 1, -2, 0, 0, 4, -2, -1, 1, 0, 5, -5, -5, 5, 5, -5, 1, -4, 0, 1, 0, 0, -6, 6, 6, 3, -2, -6, -12, 9, 6, -6, 1, -5, 1, 5, -3, -1, 1, 0, -5, 5, -1, 1, -2, 0, 0, 1, 0, 0, 0, 0, 7, -7, -7, -7, 14, 7

Offset: 1

Gus Wiseman, Nov 19 2018

Row n has length A000041(A056239(n)).
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 f(u), where h is homogeneous symmetric functions and f is forgotten symmetric functions.

			Triangle begins:
   1
   1
  -2   1
   1   0
   3  -3   1
  -3   1   0
  -4   2   4  -4   1
   1   0   0
   2   1  -2   0   0
   4  -2  -1   1   0
   5  -5  -5   5   5  -5   1
  -4   0   1   0   0
  -6   6   6   3  -2  -6 -12   9   6  -6   1
  -5   1   5  -3  -1   1   0
  -5   5  -1   1  -2   0   0
   1   0   0   0   0
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   5  -3   1   0   0   0   0
For example, row 10 gives: m(31) = 4e(4) - 2e(22) - e(31) + e(211).

Wikipedia, Symmetric polynomial

Row sums are A321747.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A321738, A321742-A321765, A321854.

cp's OEIS Frontend

A321742 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

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A321912 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in e(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

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A321728 Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.

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A321854 Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.

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A321729 Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.

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A321737 Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

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A321730 Number of ways to partition the Young diagram of an integer partition of n into vertical sections of the same sizes as the parts of the original partition.

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A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

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A321731 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections of the same sizes as the parts of the original partition.

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