A318762 -id:A318762 - cp's OEIS frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

2, 3, 10, 4, 12, 6, 5, 15, 20, 420, 7, 21, 30, 630, 120, 8, 24, 56, 840, 1680, 210, 9, 28, 60, 1980, 60060, 1260, 4324320, 11, 35, 90, 3003, 83160, 2520, 21621600, 7207200, 13, 36, 105, 7140, 180180, 5040, 24504480, 151351200, 720720

Offset: 1

Pontus von Brömssen, Sep 22 2024

Row n lists numbers m such that A376369(m) = n.
In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.
Any integer k >= 2 appears exactly once in the array.

			Array begins:
  n\k|       1         2         3         4         5          6          7          8
  ---+---------------------------------------------------------------------------------
  1  |       2         3         4         5         7          8          9         11
  2  |      10        12        15        21        24         28         35         36
  3  |       6        20        30        56        60         90        105        252
  4  |     420       630       840      1980      3003       7140       7560       9240
  5  |     120      1680     60060     83160    180180     240240     831600     900900
  6  |     210      1260      2520      5040     27720     166320    1441440    4084080
  7  | 4324320  21621600  24504480  43243200  75675600  116396280  367567200  908107200
  8  | 7207200 151351200 302702400 411863760 823727520 1816214400 2327925600 4655851200

Pontus von Brömssen, Table of n, a(n) for n = 1..1081 (antidiagonals 1..46)

Cf. A036038, A050382, A078760, A318762, A325472 (complement of first row), A325593 (complement of the union of the first 2 rows), A376369, A376376 (first column).

First five rows are A376371, A376372, A376373, A376374, A376375.

A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, 98329551, 1191578522, 15543026747, 218668538441, 3285749117475, 52700813279423, 896697825211142, 16160442591627990, 307183340680888755, 6147451460222703502, 129125045333789172825, 2841626597871149750951

Offset: 0

Simon Plouffe

This is the total number of hierarchies of n labeled elements arranged on 1 to n levels. A distribution of elements onto levels is "hierarchical" if a level l+1 contains <= elements than level l. Thus for n=4 the arrangement {1,2}:{3}{4} is not allowed. See also A140585. Examples: Let the colon ":" separate two consecutive levels l and l+1. Then n=2 --> 3: {1}{2}, {1}:{2}, {2}:{1}, n=3 --> 10: {1}{2}{3}, {1}{2}:{3}, {3}{1}:{2}, {2}{3}:{1}, {1}:{2}:{3}, {3}:{1}:{2}, {2}:{3}:{1}, {1}:{3}:{2}, {2}:{1}:{3}, {3}:{2}:{1}. - Thomas Wieder, May 17 2008
n identical objects are painted by dipping them into a long row of cans of paint of distinct colors. Begining with the first can and not skipping any cans k, 1<=k<=n, objects are dipped (painted) and not more objects are dipped into any subsequent can than were dipped into the previous can. The painted objects are then linearly ordered. - Geoffrey Critzer, Jun 08 2009
a(n) is the number of partitions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition. a(3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a. - Alois P. Heinz, Aug 30 2015
Also the number of ordered set partitions of {1,...,n} with weakly decreasing block sizes. - Gus Wiseman, Sep 03 2018
The parity of a(n) is that of A000110(A000120(n)), so a(n) is even if and only if A000120(n) == 2 (mod 3). - Álvar Ibeas, Aug 11 2020

			For n=3, say the first three cans in the row contain red, white, and blue paint respectively. The objects can be painted r,r,r or r,r,w or r,w,b and then linearly ordered in 1 + 3 + 6 = 10 ways. - _Geoffrey Critzer_, Jun 08 2009
From _Gus Wiseman_, Sep 03 2018: (Start)
The a(3) = 10 ordered set partitions with weakly decreasing block sizes:
  {{1},{2},{3}}
  {{1},{3},{2}}
  {{2},{1},{3}}
  {{2},{3},{1}}
  {{3},{1},{2}}
  {{3},{2},{1}}
  {{2,3},{1}}
  {{1,2},{3}}
  {{1,3},{2}}
  {{1,2,3}}
(End)

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. E. Hoffman, Updown categories: Generating functions and universal covers, arXiv preprint arXiv:1207.1705 [math.CO], 2012.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, 6 (1999), R2.
S. Schreiber & N. J. A. Sloane, Correspondence, 1980.

Main diagonal of: A226873, A261719, A309973.
Row sums of: A226874, A262071, A327803.
Column k=1 of A309951.
Column k=0 of A327801.
Cf. A000041, A000110, A000258, A000670, A007837, A008277, A008480, A036038, A140585, A178682, A212855, A247551, A300335, A318762.

A005651b := proc(k) add( d/(d!)^(k/d),d=numtheory[divisors](k)) ; end proc:
A005651 := proc(n) option remember; local k ; if n <= 1 then 1; else (n-1)!*add(A005651b(k)*procname(n-k)/(n-k)!, k=1..n) ; end if; end proc:
seq(A005651(k), k=0..10) ; # R. J. Mathar, Jan 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
      b(n, i-1) +binomial(n, i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015, Dec 12 2016

Table[Total[n!/Map[Function[n, Apply[Times, n! ]], IntegerPartitions[n]]], {n, 0, 20}] (* Geoffrey Critzer, Jun 08 2009 *)
Table[Total[Apply[Multinomial, IntegerPartitions[n], {1}]], {n, 0, 20}] (* Jean-François Alcover and Olivier Gérard, Sep 11 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[t==1, 1/n!, Sum[b[n-j, j, t-1]/j!, {j, i, n/t}]]; a[n_] := If[n==0, 1, n!*b[n, 0, n]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 20 2015, after Alois P. Heinz *)

a(m,n):=if n=m then 1 else sum(binomial(n,k)*a(k,n-k),k,m,(n/2))+1;
makelist(a(1,n),n,0,17); /* Vladimir Kruchinin, Sep 06 2014 */

a(n)=my(N=n!,s);forpart(x=n,s+=N/prod(i=1,#x,x[i]!));s \\ Charles R Greathouse IV, May 01 2015

{ my(n=25); Vec(serlaplace(prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n)))) } \\ Andrew Howroyd, Dec 20 2017

E.g.f.: 1 / Product (1 - x^k/k!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(-k/d). - Vladeta Jovovic, Oct 14 2002
a(n) ~ c * n!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264... . - Vaclav Kotesovec, May 09 2014
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2 , binomial(n,k)*S(n-k,k))+1, S(n,n)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 06 2014
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 1, 5, 10, 1, 1, 15, 1, 10, 15, 6, 1, 10, 10, 7, 15, 15, 1, 60, 1, 1, 21, 8, 35, 45, 1, 9, 28, 20, 1, 105, 1, 21, 105, 10, 1, 15, 35, 70, 36, 28, 1, 105, 56, 35, 45, 11, 1, 210, 1, 12, 210, 1, 84, 168, 1, 36, 55, 280, 1, 105, 1, 13, 280, 45, 126, 252, 1

Offset: 1

Max Alekseyev, Nov 07 2006

nonn

Coefficients of (D^k f)(g(t))*(D g(t))^k1*(D^2 g(t))^k2*... in the Faa di Bruno formula for D^m(f(g(t))) where k = k1 + k2 + ..., m = 1*k1 + 2*k2 + ....

Number of set partitions whose block sizes are the prime indices of n (i.e., the integer partition with Heinz number n). - Gus Wiseman, Sep 12 2018

			The a(6) = 3 set partitions of type (2,1) are {{1},{2,3}}, {{1,3},{2}}, {{1,2},{3}}. - _Gus Wiseman_, Sep 12 2018

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Faà di Bruno's Formula

Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225.

Cf. A065475, A100484.

with(numtheory):
a:= n-> (l-> add(i*l[i], i=1..nops(l))!/mul(l[i]!*i!^l[i],
         i=1..nops(l)))([seq(padic[ordp](n, ithprime(i)),
         i=1..pi(max(1, factorset(n))))]):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 14 2020

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[numSetPtnsOfType[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}] (* Gus Wiseman, Sep 12 2018 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, primepi(f[k,1])*f[k,2])!/(prod(k=1, #f~, f[k,2]!)*prod(k=1, #f~, primepi(f[k,1])!^f[k,2])); \\ Michel Marcus, Oct 11 2023

For n = p1^k1*p2^k2*... where 2 = p1 < p2 < ... are the sequence of all primes, a(n) = a([k1,k2,...]) = (k1+2*k2+...)!/((k1!*k2!*...)*(1!^k1*2!^k2*...)).

a(2*prime(n)) = n + 1, for n > 1. See A065475. - Bill McEachen, Oct 11 2023

A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 1, 2, -3, -6, 1, 3, 8, 24, -6, -120, -30, -20, 1, 720, 15, -5040, 20, 90, 144, 40320, -10, 40, -840, -15, -90, -362880, -120, 3628800, 1, -504, 5760, -420, 45, -39916800, -45360, 3360, 40, 479001600, 630, -6227020800, 504, 210, 403200, 87178291200

Offset: 1

Gus Wiseman, Sep 13 2018

sign

A refinement of Stirling numbers of the first kind.

An unsigned version is A124795.

Cf. A000041, A000110, A000258, A005651, A008480, A048994, A056239, A124794, A215366, A318762, A319182.

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[n]]-PrimeOmega[n])*numPermsOfType[primeMS[n]],{n,100}]

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (-1)^(Sum x_i * y_i - Sum y_i) (Sum x_i * y_i)! / (Product x_i^y_i * Product y_i!).

A325617 Multinomial coefficient of the prime signature of n!.

Original entry on oeis.org

1, 1, 1, 2, 4, 20, 105, 840, 3960, 51480, 675675, 10810800, 139675536, 2793510720, 58663725120, 1799020903680, 26985313555200, 782574093100800, 25992639520848000, 857757104187984000, 30021498646579440000, 1563341744336692320000, 64179292662243158400000

Offset: 0

Gus Wiseman, May 12 2019

nonn

Number of permutations of the multiset of prime factors of n!.

			The a(5) = 20 permutations of {2,2,2,3,5}:
  (22235)  (32225)  (52223)
  (22253)  (32252)  (52232)
  (22325)  (32522)  (52322)
  (22352)  (35222)  (53222)
  (22523)
  (22532)
  (23225)
  (23252)
  (23522)
  (25223)
  (25232)
  (25322)

Wikipedia, Multinomial theorem

Cf. A000142, A008480, A011371, A022559, A076934, A108731, A115627, A318762, A325272, A325273, A325508, A325543, A325544, A325609.

Table[Multinomial@@Last/@FactorInteger[n!],{n,0,15}]

a(n) = A318762(A181819(n!)).

A319225 Number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the prime indices of n.

Original entry on oeis.org

1, 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, 20, 14, 21, 13, 8, 10, 9, 18

Offset: 1

Gus Wiseman, Sep 13 2018

nonn

a(1) = 1 by convention.

A prime index of n is a number m such that prime(m) divides n.

			Of the cycle ({1,2,3}, {(1,2),(2,3),(3,1)}) the spanning subgraphs where the sizes of connected components are (2,1) are: ({1,2,3}, {(1,2)}), ({1,2,3}, {(2,3)}), ({1,2,3}, {(3,1)}). Since the prime indices of 6 are (2,1), we conclude a(6) = 3.

Wikipedia, Expressing power sums in terms of elementary symmetric polynomials
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Different orderings with signs are A115131, A210258, A263916.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A215366, A318762, A319191, A319193, A319226.

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[Length[With[{m=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},Select[Subsets[Partition[Range[Total[m]],2,1,1],{Total[m]-PrimeOmega[n]}],Sort[Length/@csm[Union[#,List/@Range[Total[m]]]]]==m&]]],{n,30}]

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

A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

Original entry on oeis.org

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

Offset: 2

Pontus von Brömssen, Sep 22 2024

nonn

a(n) is the number of occurrences of n in each of A036038, A050382, A078760, A318762, and A376367.

The sequence is unbounded. To see this, note that the sets of parts (1,1,1,4) and (2,2,3) of a partition can be exchanged without affecting the value of the multinomial coefficient, because 1+1+1+4 = 2+2+3 and 1!*1!*1!*4! = 2!*2!*3!. In particular, a((7*k)!/24^k) >= k+1 from the partitions 7*k = (3*j)*1 + j*4 + (2*(k-j))*2 + (k-j)*3 for 0 <= j <= k.

			a(6) = 3, because 6 can be written as a multinomial coefficient in 3 ways: 6 = 6!/(1!*5!) = 4!/(2!*2!) = 3!/(1!*1!*1!).

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A036038, A050382, A078760, A318762, A376367, A376368, A376370.

A319226 Irregular triangle where T(n,k) is the number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the integer partition with Heinz number A215366(n,k).

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 2, 4, 4, 1, 5, 5, 5, 5, 5, 5, 1, 6, 6, 6, 3, 2, 6, 12, 9, 6, 6, 1, 7, 7, 7, 7, 14, 7, 7, 7, 7, 7, 21, 14, 7, 7, 1, 8, 8, 8, 4, 8, 8, 8, 16, 16, 8, 2, 24, 8, 24, 12, 16, 8, 32, 20, 8, 8, 1, 9, 9, 9, 9, 9, 9, 18, 9, 9, 9, 18, 18, 3, 27, 27

Offset: 1

Gus Wiseman, Sep 13 2018

A refinement of A135278, up the sign these are the coefficients appearing in the expansion of power-sum symmetric functions in terms of elementary or homogeneous symmetric functions.

			Triangle begins:
  1
  2  1
  3  3  1
  4  2  4  4  1
  5  5  5  5  5  5  1
  6  6  6  3  2  6 12  9  6  6  1
The fourth row corresponds to the symmetric function identities:
  p(4) = -4 e(4) + 2 e(22) + 4 e(31) - 4 e(211) + e(1111)
  p(4) =  4 h(4) - 2 h(22) - 4 h(31) + 4 h(211) - h(1111).

Wikipedia, Expressing power sums in terms of elementary symmetric polynomials
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Signed versions with different row-orderings are A115131, A210258, A263916.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A215366, A318762, A319191, A319193, A319225.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[Length[Select[Subsets[Partition[Range[n],2,1,1],{n-PrimeOmega[m]}],Sort[Length/@csm[Union[#,List/@Range[n]]]]==primeMS[m]&]],{n,6},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

A332294 Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 4, 1, 6, 1, 5, 4, 8, 1, 9, 1, 8, 5, 6, 1, 12, 4, 7, 9, 10, 1, 12, 1, 16, 6, 8, 5, 18, 1, 9, 7, 16, 1, 15, 1, 12, 12, 10, 1, 24, 5, 16, 8, 14, 1, 27, 6, 20, 9, 11, 1, 24, 1, 12, 15, 32, 7, 18, 1, 16, 10, 20, 1, 36, 1, 13, 16, 18, 6

Offset: 1

Gus Wiseman, Feb 21 2020

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 sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(12) = 6 permutations:
  {1,1,2,3}
  {1,1,3,2}
  {1,2,3,1}
  {1,3,2,1}
  {2,3,1,1}
  {3,2,1,1}

MathWorld, Unimodal Sequence

Dominated by A318762.
A less interesting version is A332288.
The complement is counted by A332672.
The opposite/negative version is A332741.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Partitions whose run-lengths are unimodal are A332280.
Cf. A007052, A056239, A112798, A115981, A124010, A304660, A328509, A332283, A332578, A332638, A332671, A332742.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]],unimodQ]],{n,0,30}]

a(n) + A332672(n) = A318762(n).

a(n) = A332288(A181821(n)).

A332742 Number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 3, 2, 0, 8, 0, 3, 7, 16, 0, 24, 0, 16, 12, 4, 0, 52, 16, 5, 81, 26, 0, 54, 0, 104, 18, 6, 31, 168, 0, 7, 25, 112, 0, 99, 0, 38, 201, 8, 0, 344, 65, 132, 33, 52, 0, 612, 52, 202, 42, 9, 0, 408, 0, 10, 411, 688, 80, 162, 0, 68, 52, 272

Offset: 1

Gus Wiseman, Mar 09 2020

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 sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(n) permutations for n = 6, 8, 9, 10, 12, 14, 15, 16:
  121  132  1212  1121  1132  11121  11212  1243
       231  1221  1211  1213  11211  11221  1324
            2121        1231  12111  12112  1342
                        1312         12121  1423
                        1321         12211  1432
                        2131         21121  2143
                        2311         21211  2314
                        3121                2341
                                            2413
                                            2431
                                            3142
                                            3241
                                            3412
                                            3421
                                            4132
                                            4231

Eric Weisstein's World of Mathematics, Unimodal Sequence

Dominated by A318762.
The complement of the non-negated version is counted by A332294.
The non-negated version is A332672.
The complement is counted by A332741.
A less interesting version is A333146.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal 0-appended first differences are A332284.
Compositions whose negation is unimodal are A332578.
Partitions with non-unimodal negated run-lengths are A332639.
Numbers whose negated prime signature is not unimodal are A332642.
Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332283, A332288, A332638, A332669, A333145.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]],!unimodQ[#]&]],{n,30}]

a(n) + A332741(n) = A318762(n).

cp's OEIS Frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

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A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!*n_2!*...) where (n_1, n_2, ...) runs over all integer partitions of n.

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A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

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A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

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A325617 Multinomial coefficient of the prime signature of n!.

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A319225 Number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the prime indices of n.

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A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

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A319226 Irregular triangle where T(n,k) is the number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the integer partition with Heinz number A215366(n,k).

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A005651 Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!n_2!...) where (n_1, n_2, ...) runs over all integer partitions of n.