A226874 -id:A226874 - cp's OEIS frontend

A226884 Number of n-length words w over a 5-ary alphabet {a1,a2,...,a5} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a5) >= 1, where #(w,x) counts the letters x in word w.

120, 360, 2100, 10080, 58464, 322560, 1467840, 7155720, 36657192, 179255076, 933034830, 4330631760, 21529265392, 105723750312, 519686374512, 2614195952160, 12518753971164, 61521911547096, 302916858354682, 1486023890700624, 7449741849512920, 35973707803032480

Offset: 5

Alois P. Heinz, Jun 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A226874.

A226885 Number of n-length words w over a 6-ary alphabet {a1,a2,...,a6} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a6) >= 1, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

720, 2520, 16800, 90720, 584640, 3548160, 25098480, 125456760, 778065288, 4353689340, 26986202880, 151871712720, 970831418976, 5215235977368, 31594145929440, 180858169411920, 1087286184860568, 6173944667695728, 38303414229539712, 212004108343328400

Offset: 6

Alois P. Heinz, Jun 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A226874.

A226886 Number of n-length words w over a 7-ary alphabet {a1,a2,...,a7} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a7) >= 1, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

5040, 20160, 151200, 907200, 6431040, 42577920, 326280240, 2437475040, 15076381320, 101442781440, 685186844160, 4578510605760, 31826713309344, 215132601512160, 1519348223060640, 9879977905237440, 67264821737934744, 453057807190266432, 3094793914863208800

Offset: 7

Alois P. Heinz, Jun 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A226874.

A226887 Number of n-length words w over an 8-ary alphabet {a1,a2,...,a8} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a8) >= 1, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

40320, 181440, 1512000, 9979200, 77172480, 553512960, 4567923360, 36562125600, 322951749120, 2187661956480, 17196277250880, 125709760088640, 987050023778880, 7327777065003360, 59026901641202880, 443774894931688320, 3612457777775749056, 26312360518655164800

Offset: 8

Alois P. Heinz, Jun 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A226874.

A226888 Number of n-length words w over a 9-ary alphabet {a1,a2,...,a9} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a9) >= 1, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

362880, 1814400, 16632000, 119750400, 1003242240, 7749181440, 68518850400, 584994009600, 5490179735040, 51882551360640, 405925296678720, 3438148872412800, 28858842800988480, 242180235433825920, 2068546823359410240, 17716539920908308480, 155272065377761886400

Offset: 9

Alois P. Heinz, Jun 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 of A226874.

A226889 Number of n-length words w over a 10-ary alphabet {a1,a2,...,a10} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a10) >= 1, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

3628800, 19958400, 199584000, 1556755200, 14045391360, 116237721600, 1096301606400, 9944898163200, 98823235230720, 985768475852160, 10494386800934400, 88832292392188800, 848327839539586560, 7627235867290892160, 71992606401661397760, 656191923706230912000

Offset: 10

Alois P. Heinz, Jun 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=10 of A226874.

A318796 Number of 2n-length words w over an n-ary alphabet {a1, a2, ..., an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,an) >= 1, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 10, 180, 6496, 322560, 25098480, 2437475040, 322951749120, 51882551360640, 10494386800934400, 2503138912988313600, 720738068391525381120, 239324670990042333696000, 92995858936970165240064000, 41062460981196018797072640000, 20742554869763399771711348736000

Offset: 0

Alois P. Heinz, Sep 03 2018

nonn

			a(2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.

Alois P. Heinz, Table of n, a(n) for n = 0..238

Cf. A226874.

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:
a:= n-> `if`(n=0, 1, (2*n)!*b(2*n, 1, n)):
seq(a(n), n=0..20);

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, (2n)! b[2n, 1, n]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 05 2022, after Alois P. Heinz *)

a(n) = A226874(2n,n).

A292222 Triangle corresponding to the partition array of the M_1 multinomials (A036038).

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 10, 12, 24, 1, 15, 50, 60, 120, 1, 41, 180, 300, 360, 720, 1, 63, 497, 1260, 2100, 2520, 5040, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320, 1, 255, 5154, 20916, 58464, 90720, 151200, 181440, 362880, 1, 637, 13680, 95640, 322560, 584640, 907200, 1512000, 1814400, 3628800

Offset: 1

Wolfdieter Lang, Sep 29 2017

Abramowitz-Stegun (A-St) M_1 multinomials as partition array (partitions in A-St order) are given in A036038. See this for details.
This is the sub-triangle of A226874(n,k) for n >= k >= 1 (here k=m).
The M_1 multinomials for a partition written in exponent form P = [1^e[1], 2^e[2], ... n^e[n]] with nonnegative e[j], for j =1, ..., n, is M_1(P) = n!/Product_{j=1..n} j!^e[j]. See the A-St link.

			The triangle T(n, m) begins:
n\m  1   2     3     4      5      6      7       8       9      10 ...
1:   1
2:   1   2
3:   1   3     6
4:   1  10    12    24
5:   1  15    50    60    120
6:   1  41   180   300    360    720
7:   1  63   497  1260   2100   2520   5040
8:   1 162  1484  6496  10080  16800  20160   40320
9:   1 255  5154 20916  58464  90720 151200  181440  362880
10:  1 637 13680 95640 322560 584640 907200 1512000 1814400 3628800
...
T(5, 3) =50 because the partitions are [1^2, 3^1] and [1^1, 2^2] with M_1 numbers 20 = A036038(5, 4) and 30 = A036038(5, 5), respectively, adding to 50.

Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.

Cf. A036038, A130534 (M_2 triangle = |Stirling1|), A008277 (M_3 triangle = Stirling2), A226874 (M_1 triangle including empty partition).

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, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 29 2017, after Alois P. Heinz *)

T(n, m) = sum over the A036038 entries in row n with parts number m, for m >= n >= 1.

cp's OEIS Frontend

A226884 Number of n-length words w over a 5-ary alphabet {a1,a2,...,a5} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a5) >= 1, where #(w,x) counts the letters x in word w.

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A226885 Number of n-length words w over a 6-ary alphabet {a1,a2,...,a6} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a6) >= 1, where #(w,x) counts the letters x in word w.

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A226886 Number of n-length words w over a 7-ary alphabet {a1,a2,...,a7} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a7) >= 1, where #(w,x) counts the letters x in word w.

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A226887 Number of n-length words w over an 8-ary alphabet {a1,a2,...,a8} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a8) >= 1, where #(w,x) counts the letters x in word w.

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A226888 Number of n-length words w over a 9-ary alphabet {a1,a2,...,a9} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a9) >= 1, where #(w,x) counts the letters x in word w.

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A226889 Number of n-length words w over a 10-ary alphabet {a1,a2,...,a10} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a10) >= 1, where #(w,x) counts the letters x in word w.

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A318796 Number of 2n-length words w over an n-ary alphabet {a1, a2, ..., an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,an) >= 1, where #(w,x) counts the letters x in word w.

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A292222 Triangle corresponding to the partition array of the M_1 multinomials (A036038).

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Mathematica

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