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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. 19
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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. LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Iverson bracket Wikipedia, Multinomial coefficients Wikipedia, Partition (number theory) FORMULA T(n,k) = A226873(n,k) - [k>0] * A226873(n,k-1). EXAMPLE 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; ... MAPLE 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); MATHEMATICA 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 *) PROG (PARI) 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 CROSSREFS 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. Sequence in context: A330618 A062104 A257783 * A267901 A276561 A325746 Adjacent sequences: A226871 A226872 A226873 * A226875 A226876 A226877 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 21 2013 STATUS approved

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Last modified November 30 01:31 EST 2022. Contains 358431 sequences. (Running on oeis4.)