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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.

%I A226874 #40 Jan 19 2022 18:52:44
%S A226874 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,
%T A226874 720,0,1,63,497,1260,2100,2520,5040,0,1,162,1484,6496,10080,16800,
%U A226874 20160,40320,0,1,255,5154,20916,58464,90720,151200,181440,362880
%N 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.
%C A226874 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.
%H A226874 Alois P. Heinz, <a href="/A226874/b226874.txt">Rows n = 0..140, flattened</a>
%H A226874 Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>
%H A226874 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A226874 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A226874 T(n,k) = A226873(n,k) - [k>0] * A226873(n,k-1).
%e A226874 T(4,2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.
%e A226874 T(4,3) = 12: aabc, aacb, abac, abca, acab, acba, baac, baca, bcaa, caab, caba, cbaa.
%e A226874 T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa.
%e A226874 Triangle T(n,k) begins:
%e A226874   1;
%e A226874   0,  1;
%e A226874   0,  1,   2;
%e A226874   0,  1,   3,    6;
%e A226874   0,  1,  10,   12,   24;
%e A226874   0,  1,  15,   50,   60,   120;
%e A226874   0,  1,  41,  180,  300,   360,   720;
%e A226874   0,  1,  63,  497, 1260,  2100,  2520,  5040;
%e A226874   0,  1, 162, 1484, 6496, 10080, 16800, 20160, 40320;
%e A226874   ...
%p A226874 b:= proc(n, i, t) option remember;
%p A226874       `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
%p A226874     end:
%p A226874 T:= (n, k)-> `if`(n*k=0, `if`(n=k, 1, 0), n!*b(n, 1, k)):
%p A226874 seq(seq(T(n, k), k=0..n), n=0..12);
%p A226874 # second Maple program:
%p A226874 b:= proc(n, i) option remember; expand(
%p A226874       `if`(n=0, 1, `if`(i<1, 0, add(x^j*b(n-i*j, i-1)*
%p A226874       combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
%p A226874     end:
%p A226874 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A226874 seq(T(n), n=0..12);
%t A226874 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 *)
%o A226874 (PARI)
%o A226874 T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
%o A226874 {my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ _Andrew Howroyd_, Dec 20 2017
%Y A226874 Columns k=0-10 give: A000007, A057427, A226881, A226882, A226883, A226884, A226885, A226886, A226887, A226888, A226889.
%Y A226874 Main diagonal gives: A000142.
%Y A226874 Row sums give: A005651.
%Y A226874 T(2n,n) gives A318796.
%Y A226874 Cf. A131632, A285824, A292222, A327803.
%K A226874 nonn,tabl
%O A226874 0,6
%A A226874 _Alois P. Heinz_, Jun 21 2013