A008480 -id:A008480 - cp's OEIS frontend

A321935 Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.

1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 2, -3, 1, 6, 3, 8, 6, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 2, -1, 0, -2, 1, -6, 3, 8, -6, 1, 24, 30, 20, 15, 20, 10, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, 0, 6, -4, 3, -4, -2, 1, 4, 0, 0, -5, 0, 0, 1, -6, 0, 5, 0, 5

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).

We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is the basis of Schur functions and syt(y) is the number of standard Young tableaux of shape y.

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

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321900.

Cf. A008480, A056239, A124794, A124795, A153452 (standard Young tableaux), A215366, A296188, A300121, A319191, A319193, A321908, A321912-A321934.

A335460 Number of (1,2,1) or (2,1,2)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 8, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 20 2020

nonn

Depends only on sorted prime signature (A118914).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) compositions for n = 12, 24, 48, 36, 60, 72:
  (121)  (1121)  (11121)  (1212)  (1213)  (11212)
         (1211)  (11211)  (1221)  (1231)  (11221)
                 (12111)  (2112)  (1312)  (12112)
                          (2121)  (1321)  (12121)
                                  (2131)  (12211)
                                  (3121)  (21112)
                                          (21121)
                                          (21211)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of zeros are A303554.
The (1,2,1)-matching part is A335446.
The (2,1,2)-matching part is A335453.
Replacing "or" with "and" gives A335462.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x!=y]&]],{n,100}]

A078760 Combinations of a partition: number of ways to label a partition (of size n) with numbers 1 to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720, 1, 7, 21, 42, 35, 105, 210, 140, 210, 420, 840, 630, 1260, 2520, 5040, 1, 8, 28, 56, 56, 168, 336, 70, 280, 420, 840, 1680, 560, 1120, 1680, 3360, 6720, 2520

Offset: 0

Franklin T. Adams-Watters, Jan 08 2003

This is a function of the individual partitions of an integer. The number of values in each line is given by A000041; thus lines 0 to 5 of the sequence are (1), (1), (1,2), (1,3,6), (1,4,6,12,24). The partitions in each line are ordered with the largest part sizes first, so the line 4 indices are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1]. Note that exponents are often used to represent repeated values in a partition, so the last index could instead be written [1^4]. The combination function (sequence A007318) C(n,m) = C([m,n-m]).

This sequence is also the sequence of multinomial coefficients for partitions ordered lexicographically, matching partition sequence A080577. This is different ordering than in sequence A036038 of multinomial coefficients. - Sergei Viznyuk, Mar 15 2012

			The irregular table starts:
  [0] {1},
  [1] {1},
  [2] {1, 2},
  [3] {1, 3,  6},
  [4] {1, 4,  6, 12, 24},
  [5] {1, 5, 10, 20, 30, 60, 120},
  [6] {1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720}
  ...
C([2,1]) = 3 for the labelings ({1,2},{3}), ({1,3},{2}) and ({2,3},{2}).

T. D. Noe, Rows n=0..25 of triangle, flattened
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arxiv:1405.5283 [math.NT], 2014.
Sergei Viznyuk, C Program
Index entries for triangles and arrays related to Pascal's triangle.

Different from A036038.

Cf. A000041, A008480, A063008, A080577.

g:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 25 2020

Flatten[Table[Apply[Multinomial, IntegerPartitions[i], {1}], {i,0,25}]] (* T. D. Noe, Oct 14 2007 *)
Flatten[ Multinomial @@@ IntegerPartitions @ # & /@ Range[ 0, 8]] (* Michael Somos, Feb 05 2011 *)
g[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times@@(ee!)];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[ Prepend[#, i] & /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
row[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}] & /@ b[n, n];
T[n_] := g /@ row[n];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

C(sig)={vecsum(sig)!/vecprod(apply(k->k!, sig))}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) }  \\ Andrew Howroyd, Mar 25 2020

def A070289_row(n): return [multinomial(x) for x in Partitions(n)]
print(flatten([A070289_row(n) for n in range(8)]))  # Peter Luschny, Jun 24 2025

C([]) = (Sum a(i))! / Product a(i) !.

T(n,k) = A008480(A063008(n,k)). - Andrew Howroyd, Mar 25 2020

A335462 Number of (1,2,1) and (2,1,2)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 20 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 36, 72, 270, 144, 300:
  (1,2,1,2)  (1,1,2,1,2)  (2,1,2,3,2)  (1,1,1,2,1,2)  (1,2,3,1,3)
  (2,1,2,1)  (1,2,1,1,2)  (2,1,3,2,2)  (1,1,2,1,1,2)  (1,3,1,2,3)
             (1,2,1,2,1)  (2,2,1,3,2)  (1,1,2,1,2,1)  (1,3,1,3,2)
             (2,1,1,2,1)  (2,2,3,1,2)  (1,2,1,1,1,2)  (1,3,2,1,3)
             (2,1,2,1,1)  (2,3,1,2,2)  (1,2,1,1,2,1)  (1,3,2,3,1)
                          (2,3,2,1,2)  (1,2,1,2,1,1)  (2,1,3,1,3)
                                       (2,1,1,1,2,1)  (2,3,1,3,1)
                                       (2,1,1,2,1,1)  (3,1,2,1,3)
                                       (2,1,2,1,1,1)  (3,1,2,3,1)
                                                      (3,1,3,1,2)
                                                      (3,1,3,2,1)
                                                      (3,2,1,3,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The avoiding version is A333175.
Replacing "and" with "or" gives A335460.
Positions of nonzero terms are A335463.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Dimensions of downsets of standard compositions are A335465.
Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x_,x_,_,y_,_,x_,_}/;x>y]&]],{n,100}]

A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

Original entry on oeis.org

36, 72, 90, 100, 108, 126, 144, 180, 196, 198, 200, 216, 225, 234, 252, 270, 288, 300, 306, 324, 342, 350, 360, 378, 392, 396, 400, 414, 432, 441, 450, 468, 484, 500, 504, 522, 525, 540, 550, 558, 576, 588, 594, 600, 612, 630, 648, 650, 666, 675, 676, 684, 700

Offset: 1

Gus Wiseman, Jun 20 2020

nonn

A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   72: {1,1,1,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  126: {1,2,2,4}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  196: {1,1,4,4}
  198: {1,2,2,5}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  234: {1,2,2,6}
  252: {1,1,2,2,4}
  270: {1,2,2,2,3}
  288: {1,1,1,1,1,2,2}
  300: {1,1,2,3,3}

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Replacing "and" with "or" gives A126706.
Positions of nonzero terms in A335462.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Cf. A056239, A056986, A112798, A158005, A181796, A333175, A335451, A335452, A335460, A335465.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Permutations[primeMS[#]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x_,x_,_,y_,_,x_,_}/;x>y]&]!={}&]

A321648 Number of permutations of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 3, 1, 1, 2, 1, 3, 6, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 4, 2, 1, 8, 15, 3, 1, 12, 1, 5, 3, 9, 1, 2, 1, 3, 21, 6, 1, 2, 10, 4, 28, 10, 1, 6, 1, 11, 6, 1, 20, 20, 1, 7, 36, 12, 1, 2, 1, 12, 3, 8, 5, 30, 1, 3, 1, 13

Offset: 1

Gus Wiseman, Nov 15 2018

nonn

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

			The a(42) = 12 permutations: (3211), (3121), (3112), (2311), (2131), (2113), (1321), (1312), (1231), (1213), (1132), (1123).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A008480, A056239, A112798, A122111, A296150, A321645, A321646, A321647, A321649, A321650.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Permutations[conj[primeMS[n]]]],{n,50}]

A008480(n) = {my(sig=factor(n)[, 2]); vecsum(sig)!/factorback(apply(k->k!, sig))}; \\ From A008480
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A321648(n) = A008480(A122111(n)); \\ Antti Karttunen, Dec 23 2018

a(n) = A008480(A122111(n)).

A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 15 2021

nonn

First differs from A348383 at a(216) = 27, A348383(216) = 28.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
These permutations are ordered factorizations of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
The version without twins for n > 0 is a(n) + 1 if n is a perfect square; otherwise a(n).

			The factorizations for n = 4, 12, 24, 30, 36, 48, 60, 64, 72:
  4    12     24     30     36       48       60       64       72
  2*2  2*6    3*8    5*6    4*9      6*8      2*30     8*8      8*9
       3*4    4*6    2*15   6*6      2*24     3*20     2*32     2*36
       2*2*3  2*12   3*10   2*18     3*16     4*15     4*16     3*24
              2*2*6  2*3*5  3*12     4*12     5*12     2*4*8    4*18
              2*3*4         2*2*9    2*3*8    6*10     2*2*16   6*12
                            2*3*6    2*4*6    2*5*6    2*2*4*4  2*4*9
                            3*3*4    3*4*4    3*4*5             2*6*6
                            2*2*3*3  2*2*12   2*2*15            3*3*8
                                     2*2*3*4  2*3*10            3*4*6
                                              2*2*3*5           2*2*18
                                                                2*3*12
                                                                2*2*3*6
                                                                2*3*3*4
                                                                2*2*2*3*3
The a(270) = 19 factorizations:
  (2*3*5*9)   (5*6*9)   (3*90)   (270)
  (3*3*5*6)   (2*3*45)  (5*54)
  (2*3*3*15)  (2*5*27)  (6*45)
              (2*9*15)  (9*30)
              (3*3*30)  (10*27)
              (3*5*18)  (15*18)
              (3*6*15)  (2*135)
              (3*9*10)
Note that (2*3*3*3*5) is separable but has no alternating permutations.

Partitions not of this type are counted by A344654, ranked by A344653.
Partitions of this type are counted by A344740, ranked by A344742.
The complement is counted by A347706, without twins A348380.
The case without twins is A348379.
Dominates A348383, the separable case.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A049774, A102726, A119620, A128761, A344614, A347437, A347438, A347458, A348381.

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],Function[f,Select[Permutations[f],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]!={}]]],{n,100}]

For n > 1, a(n) = A335434(n) + A010052(n).

A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 02 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.

A prime index of n is a number m such that prime(m) divides n. For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

			The following are the weakly alternating permutations for selected n:
n = 2   6    12    24     48      60     90     120     180
   ----------------------------------------------------------
    2   23   223   2223   22223   2253   2335   22253   22335
        32   232   2232   22232   2325   2533   22325   22533
             322   2322   22322   2523   3253   22523   23253
                   3222   23222   3252   3325   23252   23352
                          32222   3522   3352   25232   25233
                                  5232   3523   32225   25332
                                         5233   32522   32325
                                         5332   35222   32523
                                                52223   33252
                                                52322   33522
                                                        35232
                                                        52323
                                                        53322

Counting all permutations of prime factors gives A008480.
The variation counting anti-run permutations is A335452.
The strong case is A345164, with twins A344606.
Compositions of this type are counted by A349052, also A129852 and A129853.
Compositions not of this type are counted by A349053, ranked by A349057.
The version for patterns is A349058, strong A345194.
The version for ordered factorizations is A349059, strong A348610.
Partitions of this type are counted by A349060, complement A349061.
The complement is counted by A349797.
The non-alternating case is A349798.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A071321 gives the alternating sum of prime factors, reverse A071322.
A344616 gives the alternating sum of prime indices, reverse A316524.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A348379 counts factorizations with an alternating permutation.
A349800 counts weakly but not strongly alternating compositions.
Cf. A028234, A051119, A096441, A335433, A335448, A344614, A344652, A344653, A345173, A345192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Permutations[primeMS[n]],whkQ[#]||whkQ[-#]&]],{n,100}]

A096825 Maximal size of an antichain in divisor lattice D(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com) and Vladeta Jovovic, Aug 17 2004

nonn

The divisor lattice D(n) is the lattice of the divisors of the natural number n.

Also the number of divisors of n with half (rounded either way) as many prime factors (counting multiplicity) as n. - Gus Wiseman, Aug 24 2018

			There are two maximal size antichains of divisors of 180, namely {12, 18, 20, 30, 45} and {4, 6, 9, 10, 15}. Both have length 5 so a(180) = 5. - _Gus Wiseman_, Aug 24 2018

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
S.-H. Cha, E. G. DuCasse and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT] (2014), (2.19).

Cf. A001055, A008480, A096826, A096827, A253249, A285573.

a:=proc(n) local klist,x; klist:=ifactors(n)[2,1..-1,2]; coeff(normal(mul((1-x^(k+1))/(1-x),k=klist)),x,floor(add(k,k=klist)/2)) end: seq(a(n), n=1..100);

a[n_] := Module[{pp, kk, x}, {pp, kk} = Transpose[FactorInteger[n]]; Coefficient[ Product[ Total[x^Range[0, k]], {k, kk}], x, Quotient[ Total[ kk], 2] ] ]; Array[a, 100] (* Jean-François Alcover, Nov 20 2017 *)
Table[Length[Select[Divisors[n],PrimeOmega[#]==Round[PrimeOmega[n]/2]&]],{n,50}] (* Gus Wiseman, Aug 24 2018 *)

a(n)=if(n<6||isprimepower(n), return(1)); my(d=divisors(n),r=1,u); d=d[2..#d-1];for(k=0,2^#d-1,if(hammingweight(k)<=r,next); u=vecextract(d,k); for(i=1,#u, for(j=i+1,#u, if(u[j]%u[i]==0, next(3))));r=#u);r \\ Charles R Greathouse IV, May 14 2013

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A096825(n):
    fs = factorint(n)
    return len(list(multiset_combinations(fs,sum(fs.values())//2))) # Chai Wah Wu, Aug 23 2021

def A096825(n) :
    if n==1 : return 1
    R. = QQ[]; mults = [x[1] for x in factor(n)]
    return prod((t^(m+1)-1)//(t-1) for m in mults)[sum(mults)//2]
# Eric M. Schmidt, May 11 2013

a(n) is the coefficient at x^k in (1+x+...+x^k_1)*...*(1+x+...+x^k_q) where n=p_1^k_1*...*p_q^k_q is the prime factorization of n and k=floor((k_1+...+k_q)/2). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 22 2004

More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 22 2004

A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1

Offset: 2

Matthew Vandermast, Jun 03 2012

Length of row n equals A001221(n).
The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences.  Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.
(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)
Table lists exponents in the order in which they appear in the prime factorization of a member of A025487.  This ordering is common in database comments (e.g., A008966).
Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.
Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022

			First rows of table read:
  1;
  1;
  2;
  1;
  1,1;
  1;
  3;
  2;
  1,1;
  1;
  2,1;
  ...
The multiset of positive exponents in the prime factorization of 6 = 2*3 is {1,1} (1s are often left implicit as exponents). The prime signature of 6 is therefore {1,1}.
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, as does 18 = 2*3^2. Rows 12 and 18 of the table both read {2,1}.

Jason Kimberley, Table of i, a(i) for i = 2..24301 (n = 2..10000)

Cf. A025487, A001221 (row lengths), A001222 (row sums). A118914 gives the nondecreasing version. A124010 lists exponents in n's prime factorization in natural order, with A124010(1) = 0.
A212172 cross-references over 20 sequences that depend solely on n's prime exponents >= 2, including the "core" sequence A000688.  Other sequences determined by the exponents in the prime factorization of n include:
Multiplicative: A000005, A007425, A008683, A008836, A034444, A037445, A181819.
Additive: A001221, A001222, A056169.
Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
A highly incomplete selection of sequences, each definable by the set of prime signatures possessed by its members: A000040, A000290, A000578, A000583, A000961, A001248, A001358, A001597, A001694, A002808, A004709, A005117, A006881, A013929, A030059, A030229, A052486.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // Jason Kimberley, Jun 13 2012

apply( {A212171_row(n)=vecsort(factor(n)[,2]~,,4)}, [1..40])\\ M. F. Hasler, Apr 19 2022

Row n of A118914, reversed.

Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.

cp's OEIS Frontend

A321935 Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.

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A335460 Number of (1,2,1) or (2,1,2)-matching permutations of the prime indices of n.

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A078760 Combinations of a partition: number of ways to label a partition (of size n) with numbers 1 to n.

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A335462 Number of (1,2,1) and (2,1,2)-matching permutations of the prime indices of n.

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A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

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A321648 Number of permutations of the conjugate of the integer partition with Heinz number n.

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A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

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A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

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