A376661 -id:A376661 - cp's OEIS frontend

A070289 Number of distinct values of multinomial coefficients ( n / (p1, p2, p3, ...) ) where (p1, p2, p3, ...) runs over all partitions of n.

1, 1, 2, 3, 5, 7, 11, 14, 20, 27, 36, 47, 64, 79, 102, 125, 157, 193, 243, 296, 366, 441, 538, 639, 773, 911, 1092, 1294, 1532, 1799, 2131, 2475, 2901, 3369, 3935, 4554, 5292, 6084, 7033, 8087, 9292, 10617, 12198, 13880, 15874, 18039, 20541, 23263, 26414, 29838

Offset: 0

Naohiro Nomoto, May 12 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..92
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, Journal of Number Theory 118 (2006), 15-30; arXiv preprint, arXiv:math/0509470 [math.CO], 2005.
Sergei Viznyuk, C-Program, C-Program, local copy.

Cf. A000041, A036038, A212855, A215520, A215521, A309951, A309999, A325305, A376661.

b:= proc(n,i) option remember;
      if n=0 then {1} elif i<1 then {} else {b(n, i-1)[],
         seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)} fi
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 14 2012

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Union[Join[b[n, i-1], Flatten[ Table[Function[{x}, x*i!^j] /@ b[n-i*j, i-1], {j, 1, n/i}]]]]]]; a[n_] := Length[b[n, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)

def A070289(n):
    P = Partitions(n)
    M = set(multinomial(list(x)) for x in P)
    return len(M)
[A070289(n) for n in range(20)]
# Joerg Arndt, Aug 14 2012

a(n) = A215520(n,n) = A215521(2*n,n). - Alois P. Heinz, Nov 08 2012

Terms a(n) for n >= 45 corrected by Joerg Arndt and Alois P. Heinz, Aug 14 2012

A376663 Largest frequency of n in the multiset of multinomial coefficients k!/(x_1! * ... * x_j!) with 1 <= x_1 <= ... <= x_j for a fixed k = x_1 + ... + x_j, i.e., maximum number of times that n appears in a row of A036038.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			56 appears twice in row 8 of A036038 (and never more than twice in the same row): 56 = 8!/(1!*1!*6!) = 8!/(3!*5!). Hence, a(56) = 2.

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

Cf. A036038, A376369, A376661, A376664, A376665 (records), A376666 (indices of records), A376667.

A376662 The smallest of the most common numbers among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 210, 56, 504, 1260, 9240, 166320, 360360, 5045040, 75675600, 1210809600, 4084080, 73513440, 698377680, 13967553600, 146659312800, 1075501627200, 37104806138400, 296838449107200, 7420961227680000, 96472495959840000, 2604757390915680000

Offset: 0

Pontus von Brömssen, Oct 02 2024

nonn

a(n) is the smallest number that appears A376661(n) times in row n of A036038 (for n >= 1) or A078760.

			For n = 8, the only numbers that appear more than once in row 8 of A036038 are 56 and 1680, which both appear twice. Since 56 < 1680, a(8) = 56.

Cf. A036038, A078760, A376661.

A376821 Number of irreducible pairs of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 3, 3, 6, 7, 8, 4, 6, 7, 12, 17, 23, 23, 31, 38, 36, 70, 71, 101, 127, 118, 145, 191, 209, 261, 309, 396, 462, 512, 652, 769, 878, 1097, 1320, 1563, 1827, 2098, 2533, 2932, 3475, 4185, 4756, 5726, 6614, 7686, 9189, 10825

Offset: 0

Pontus von Brömssen, Oct 05 2024

nonn

A pair of partitions of n is irreducible if the two partitions yield the same multinomial coefficient but have no parts in common. The partitions in the pair are required to be distinct, otherwise a(0) would be 1.

			   n | irreducible pairs of partitions of n
  ---+-------------------------------------
   7 | (1,1,1,4), (2,2,3)
   8 | (1,1,6), (3,5)
  10 | (1,4,5), (2,2,6)
  13 | (1,1,1,10), (6,7)
     | (1,1,3,8), (2,4,7)
     | (1,1,1,1,1,8), (2,2,2,7)
  14 | (1,2,2,9), (3,3,8)
     | (1,1,1,2,9), (3,4,7)
     | (1,1,1,1,1,1,4,4), (2,2,2,2,3,3)

George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, Journal of Number Theory 118 (2006), 15-30; arXiv preprint, arXiv:math/0509470 [math.CO], 2005. (See Section 7.)

Cf. A070289, A260669, A325306, A376661.

A376828 a(n) = smallest integer k >= 0 such that there exist n disjoint partitions of k that yield the same multinomial coefficient.

Original entry on oeis.org

0, 7, 31, 50

Offset: 1

Pontus von Brömssen, Oct 05 2024

a(5) > 153 if it exists. - Max Alekseyev, May 18 2025

			a(2) = 7 is the smallest k such that A376821(k) > 0.
    |       | n partitions of a(n) with a common multinomial coefficient
  n |  a(n) |  but no part appearing in more than one of the partitions
  --+-------+-----------------------------------------------------------
  1 |    0  | ()
  2 |    7  | (1,1,1,4), (2,2,3)
  3 |   31  | (1,1,4,4,4,7,10), (2,2,2,2,5,9,9), (3,3,3,6,8,8)
  4 |   50  | (1,1,1,9,12,26), (2,2,8,11,27), (3,3,3,16,25), (5,6,15,24)

Cf. A376661, A376673, A376821, A376866.

Name edited by Max Alekseyev, May 18 2025

cp's OEIS Frontend

A070289 Number of distinct values of multinomial coefficients ( n / (p1, p2, p3, ...) ) where (p1, p2, p3, ...) runs over all partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A376663 Largest frequency of n in the multiset of multinomial coefficients k!/(x_1! * ... * x_j!) with 1 <= x_1 <= ... <= x_j for a fixed k = x_1 + ... + x_j, i.e., maximum number of times that n appears in a row of A036038.

Views

Author

Keywords

Examples

Links

Crossrefs

A376662 The smallest of the most common numbers among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A376821 Number of irreducible pairs of partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A376828 a(n) = smallest integer k >= 0 such that there exist n disjoint partitions of k that yield the same multinomial coefficient.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions