A318762 -id:A318762 - cp's OEIS frontend

A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

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

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386578 with rows reversed.
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}.

			Row n = 21 counts the following permutations:
  .  111122  111221  111212  112121  .
     221111  112211  112112  121121
             122111  121112  121211
             211112  211121
                     211211
                     212111
Triangle begins:
  .
  1
  1  0
  0  2
  1  0  0
  0  2  1
  1  0  0  0
  0  0  6
  0  2  2  2
  0  2  2  0
  1  0  0  0  0
  0  0  6  6
  1  0  0  0  0  0
  0  2  3  0  0
  0  2  3  4  1
  0  0  0 24
  1  0  0  0  0  0  0
  0  0  6 12 12
  1  0  0  0  0  0  0  0
  0  0  6 12  2
  0  2  4  6  3  0

Column k = 0 is A010051.
Row lengths are A056239.
Row sums are A318762.
Column k = last is A335125.
For prime indices we have A374252, reverse A386577.
Reversing all rows gives A386578.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A305936 is a multiset whose multiplicities are the prime indices of n.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A001222, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A386581.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
ugt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]!=q[[#+1]]&]]==x]];
Table[Table[Length[ugt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]

A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 12, 8, 28, 20, 32, 38, 112, 76, 116, 58, 352, 236, 1296, 176, 540, 288, 4448, 374, 612, 1144, 1812, 824, 16640, 1316, 59968, 612, 2336, 4528, 3208, 2924, 231168, 18320, 10632, 2168, 856960, 7132, 3334400, 3776, 11684, 74080, 12679424, 4919, 19192

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset 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 tree-partition of m is either m itself or a sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts.

			The a(6) = 6 tree-partitions of {1,1,2}:
  (112)
  ((1)(12))
  ((2)(11))
  ((1)(1)(2))
  ((1)((1)(2)))
  ((2)((1)(1)))

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318848.

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}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[allmsptrees[nrmptn[n]]],{n,20}]

a(n) = A281118(A181821(n)).
a(prime(n)) = A289501(n).
a(2^n) = A005804(n).

More terms from Jinyuan Wang, Jun 26 2020

A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386579 with rows reversed.
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}.

			Row n = 21 counts the following permutations:
  .  112121  111212  111221  111122  .
     121121  112112  112211  221111
     121211  121112  122111
             211121  211112
             211211
             212111
Triangle begins
   .
   1
   0  1
   2  0
   0  0  1
   1  2  0
   0  0  0  1
   6  0  0
   2  2  2  0
   0  2  2  0
   0  0  0  0  1
   6  6  0  0
   0  0  0  0  0  1
   0  0  3  2  0
   1  4  3  2  0
  24  0  0  0
   0  0  0  0  0  0  1
  12 12  6  0  0
   0  0  0  0  0  0  0  1
   2 12  6  0  0
   0  3  6  4  2  0

Column k = last is A010051.
Row lengths are A056239.
Initial zeros are counted by A252736 = A001222 - 1.
Row sums are A318762.
Column k = 0 is A335125.
For prime indices we have A386577.
Reversing all rows gives A386579.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A305936 is a multiset whose multiplicities are the prime indices of n.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A374246, A382525, A386581.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aqt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==x]];
Table[Table[Length[aqt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]

A318808 Number of Lyndon permutations of a multiset whose multiplicities are the prime indices of n > 1.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 2, 6, 0, 6, 0, 4, 2, 1, 0, 12, 3, 1, 14, 5, 0, 10, 0, 24, 3, 1, 5, 30, 0, 1, 3, 20, 0, 15, 0, 6, 30, 1, 0, 60, 8, 20, 4, 7, 0, 90, 7, 30, 4, 1, 0, 60, 0, 1, 51, 120, 9, 21, 0, 8, 5, 35, 0, 180, 0, 1, 70, 9, 14, 28, 0, 120

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset 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 Lyndon product of two or more finite sequences is defined to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product.
a(1) = 1 by convention.

			The a(30) = 10 Lyndon permutations of {1,1,1,2,2,3}:
  (111223)
  (111232)
  (111322)
  (112123)
  (112132)
  (112213)
  (112312)
  (113122)
  (113212)
  (121213)

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Wikipedia, Lyndon word

Cf. A000670, A005651, A008480, A019536, A056239, A060223, A112798, A296372, A298941, A305936, A318762, A318810.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Permutations[nrmptn[n]],LyndonQ]],{n,2,100}]

sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
count(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, moebius(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
a(n)={if(n==1, 1, count(sig(n)))} \\ Andrew Howroyd, Dec 08 2018

a(p) = 0 for prime p. - Andrew Howroyd, Dec 08 2018

A376379 Heinz numbers of integer partitions (x_1, ..., x_k) with at least 2 parts, sorted by increasing multinomial coefficients (x_1 + ... + x_k)!/(x_1! * ... * x_k!). In case of ties, the partitions are sorted in standard order as in A080577.

Original entry on oeis.org

4, 6, 10, 14, 8, 9, 22, 26, 34, 38, 15, 46, 58, 12, 62, 74, 82, 21, 86, 94, 106, 118, 122, 20, 25, 134, 33, 142, 146, 158, 16, 166, 178, 194, 202, 39, 206, 214, 18, 28, 218, 226, 254, 262, 274, 35, 278, 51, 298, 302, 314, 326, 334, 346, 44, 358, 362, 382, 57

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

This is a permutation of the composite numbers A002808.

			  n | A376367(n) | partition | a(n)
  --+------------+-----------+-----
  1 |     2      |  (1,1)    |   4
  2 |     3      |  (2,1)    |   6
  3 |     4      |  (3,1)    |  10
  4 |     5      |  (4,1)    |  14
  5 |     6      |  (1,1,1)  |   8
  6 |     6      |  (2,2)    |   9
  7 |     6      |  (5,1)    |  22
The number 210 appears 6 times in A376367, corresponding to the partitions (4,1,1,1), (3,2,2), (6,4), (13,1,1), (19,2), and (209,1), with Heinz numbers 56, 45, 91, 164, 201 and 2578, respectively. These numbers appear as a(257), ..., a(262).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A002808, A080577, A318762, A376367.

A318762(a(n)) = A376367(n).

A319192 Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -6, 3, 8, -6, 1, 24, -30, -20, 15, 20, -10, 1, -120, 90, 144, 40, -15, -90, -120, 45, 40, -15, 1, 720, -840, -504, -420, 630, 504, 210, 280, -105, -210, -420, 105, 70, -21, 1, -5040, 5760, 3360, 1260, -3360, 2688, -1260, -4032, -3360, -1120

Offset: 1

Gus Wiseman, Sep 13 2018

A generalization of the triangle of Stirling numbers of the first kind, these are the coefficients appearing in the expansion of single-part augmented elementary symmetric functions in terms of power-sum symmetric functions.

			Triangle begins:
   1
  -1   1
   2  -3   1
  -6   3   8  -6   1
  24 -30 -20  15  20 -10   1
The fourth row corresponds to the symmetric function identity: 24 e(4) = -6 p(4) + 3 p(22) + 8 p(31) - 6 p(211) + p(1111).

Other row orderings are A036039 and A102189.

Cf. A000041, A005651, A008277, A008480, A048994, A056239, A124794, A124795, A215366, A318762, A319182, A319191.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*numPermsOfType[primeMS[m]],{n,5},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

A330415 Coefficient of h(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where h is the basis of homogeneous symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 2, -1, 3, -3, 4, 1, -2, -4, 5, 4, 6, -5, -5, -1, 7, 5, 8, 5, -6, -6, 9, -5, -3, -7, 2, 6, 10, 12, 11, 1, -7, -8, -7, -9, 12, -9, -8, -6, 13, 14, 14, 7, 7, -10, 15, 6, -4, 7, -9, 8, 16, -7, -8, -7, -10, -11, 17, -21, 18, -12, 8, -1, -9, 16, 19, 9, -11, 16

Offset: 1

Gus Wiseman, Dec 14 2019

sign

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

Up to sign, a(n) is the number of acyclic spanning subgraphs of an undirected n-cycle whose component sizes are the prime indices of n.

The unsigned version (except with a(1) = 1) is A319225.
The transition from p to e by Heinz numbers is A321752.
The transition from p to h by Heinz numbers is A321754.
Different orderings with and without signs and first terms are A115131, A210258, A263916, A319226, A330417.
Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.

Table[If[n==1,0,(-1)^(PrimeOmega[n]-1)*Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]*(PrimeOmega[n]-1)!/(Times@@Factorial/@FactorInteger[n][[All,2]])],{n,30}]

a(n) = (-1)^(Omega(n) - 1) * A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.

A330417 Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, -2, 1, 3, -3, -4, 1, 2, 4, 5, -4, -6, -5, -5, 1, 7, 5, -8, 5, 6, 6, 9, -5, 3, -7, -2, -6, -10, -12, 11, 1, -7, 8, -7, 9, -12, -9, 8, 6, 13, 14, -14, 7, 7, 10, 15, -6, 4, 7, -9, -8, -16, -7, 8, -7, 10, -11, 17, -21, -18, 12, -8, 1, -9, -16, 19, 9, -11

Offset: 1

Gus Wiseman, Dec 14 2019

sign

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

Up to sign, a(n) is the number of acyclic spanning subgraphs of an undirected n-cycle whose component sizes are the prime indices of n.

The unsigned version (except with a(1) = 1) is A319225.
The transition from p to e by Heinz numbers is A321752.
The transition from p to h by Heinz numbers is A321754.
Different orderings with and without signs and first terms are A115131, A210258, A263916, A319226, A330415.
Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.

Table[If[n==1,0,With[{tot=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]},(-1)^(tot-PrimeOmega[n])*tot*(PrimeOmega[n]-1)!/(Times@@Factorial/@FactorInteger[n][[All,2]])]],{n,30}]

a(n) = (-1)^(A056239(n) - Omega(n)) * A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.

cp's OEIS Frontend

A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

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A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

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A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

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A318808 Number of Lyndon permutations of a multiset whose multiplicities are the prime indices of n > 1.

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A376379 Heinz numbers of integer partitions (x_1, ..., x_k) with at least 2 parts, sorted by increasing multinomial coefficients (x_1 + ... + x_k)!/(x_1! * ... * x_k!). In case of ties, the partitions are sorted in standard order as in A080577.

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A319192 Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

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A330415 Coefficient of h(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where h is the basis of homogeneous symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

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A330417 Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

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