A226883 -id:A226883 - cp's OEIS frontend

A226874 Number T(n,k) of n-length words w over a k-ary alphabet {a1, a2, ..., ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 10, 12, 24, 0, 1, 15, 50, 60, 120, 0, 1, 41, 180, 300, 360, 720, 0, 1, 63, 497, 1260, 2100, 2520, 5040, 0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320, 0, 1, 255, 5154, 20916, 58464, 90720, 151200, 181440, 362880

Offset: 0

Alois P. Heinz, Jun 21 2013

T(n,k) is the sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a multiset of size k.

			T(4,2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.
T(4,3) = 12: aabc, aacb, abac, abca, acab, acba, baac, baca, bcaa, caab, caba, cbaa.
T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   2;
  0,  1,   3,    6;
  0,  1,  10,   12,   24;
  0,  1,  15,   50,   60,   120;
  0,  1,  41,  180,  300,   360,   720;
  0,  1,  63,  497, 1260,  2100,  2520,  5040;
  0,  1, 162, 1484, 6496, 10080, 16800, 20160, 40320;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Iverson bracket
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Columns k=0-10 give: A000007, A057427, A226881, A226882, A226883, A226884, A226885, A226886, A226887, A226888, A226889.
Main diagonal gives: A000142.
Row sums give: A005651.
T(2n,n) gives A318796.
Cf. A131632, A285824, A292222, A327803.

b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
T:= (n, k)-> `if`(n*k=0, `if`(n=k, 1, 0), n!*b(n, 1, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(x^j*b(n-i*j, i-1)*
      combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);

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}]]; t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from first Maple *)

T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
{my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ Andrew Howroyd, Dec 20 2017

T(n,k) = A226873(n,k) - [k>0] * A226873(n,k-1).

A214309 a(n) is the number of representative four-color bracelets (necklaces with turning over allowed) with n beads, for n >= 4.

Original entry on oeis.org

3, 6, 26, 93, 424, 1180, 4844, 16165, 66953, 216804, 852822, 2949804, 12119134, 40886724, 160826008, 572457489, 2331396595, 8104270828, 32043699894, 115995102806, 471268872328, 1674576998468, 6641876380417, 24364816845446, 98894256728960, 357006263815751

Offset: 4

Wolfdieter Lang, Jul 31 2012

nonn

This is the fourth column (m=4) of triangle A213940.
The relevant p(n,4)= A008284(n,4) representative color multinomials have exponents (signatures) from the 4 part partitions of n, written with nonincreasing parts. E.g., n=6: [3,1,1,1] and [2,2,1,1] (p(6,4)=2). The corresponding representative bracelets have the four-color multinomials c[1]^3*c[2]*c[3]*c[4] and c[1]^2*c[2]^2*c[3]*c[4].
Compare this with A032275 where also bracelets with less than four colors are included, and not only representatives are counted.
Number of n-length bracelets w over a 4-ary alphabet {a1,a2,...,a4} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a4) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226883). The number of 4 color bracelets up to permutations of colors is given by A056359. - Andrew Howroyd, Sep 26 2017

			a(4) = A213939(4,5) = 3 from the representative bracelets (with colors  j for c[j], j=1, 2, ..., 4) 1234, 1342 and 1423, all taken cyclically. The necklace cyclic(1324), for example, becomes equivalent to cyclic(1423) under the dihedral D_4 turning over (or reflection) operation.
a(6) = A213939(6, 8) = A213939(6, 9) =  10 + 16 = 26. See the comment above for the representative color multinomials for each case.

Andrew Howroyd, Table of n, a(n) for n = 4..200

Cf. A213939, A213940, A214311 (m=5), A214312 (m=4, all bracelets).

Cf. A056359, A226883.

Cyc(v)={my(g=fold(gcd,v), s=vecsum(v)); sumdiv(g, d, eulerphi(d)*(s/d)!/prod(i=1, #v, (v[i]/d)!))/s}
CPal(v)={my(odds=#select(t->t%2,v), s=vecsum(v));  if(odds>2, 0, ((s-odds)/2)!/prod(i=1, #v, (v[i]\2)!))}
a(n)={my(t=0); forpart(p=n, t+=Cyc(Vec(p))+CPal(Vec(p)), [1,n], [4,4]); t/2} \\ Andrew Howroyd, Sep 26 2017

a(n) = A213940(n,4), n >= 4.

a(n) = sum(A213939(n,k),k=(2+floor(n/2) + p(n,3))..(p(n,4)+1+floor(n/2)+p(n,3))), n>=4, with p(n,m) = A008284(n,m) the number of partitions of n with m parts.

Terms a(26) and beyond from Andrew Howroyd, Sep 26 2017

cp's OEIS Frontend

A226874 Number T(n,k) of n-length words w over a k-ary alphabet {a1, a2, ..., ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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