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 A213275 Number A(n,k) of words w where each letter of the k-ary alphabet occurs n times and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z; square array A(n,k), n>=0, k>=0, read by antidiagonals. 19
 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 15, 7, 1, 1, 1, 120, 105, 106, 19, 1, 1, 1, 720, 945, 2575, 1075, 56, 1, 1, 1, 5040, 10395, 87595, 115955, 13326, 174, 1, 1, 1, 40320, 135135, 3864040, 19558470, 7364321, 188196, 561, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The words counted by A(n,k) have length n*k. LINKS Alois P. Heinz, Antidiagonals n = 0..20, flattened EXAMPLE A(0,k) = A(n,0) = 1: the empty word. A(n,1) = 1: (a)^n for alphabet {a}. A(1,2) = 2: ab, ba for alphabet {a,b}. A(1,3) = 6: abc, acb, bac, bca, cab, cba for alphabet {a,b,c}. A(2,2) = 3: aabb, abab, baab. A(2,3) = 15: aabbcc, aabcbc, aacbbc, ababcc, abacbc, abcabc, acabbc, acbabc, baabcc, baacbc, bacabc, bcaabc, caabbc, cababc, cbaabc. A(3,2) = 7: aaabbb, aababb, aabbab, abaabb, ababab, baaabb, baabab. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 2, 6, 24, 120, 720, ... 1, 1, 3, 15, 105, 945, 10395, ... 1, 1, 7, 106, 2575, 87595, 3864040, ... 1, 1, 19, 1075, 115955, 19558470, 4622269345, ... 1, 1, 56, 13326, 7364321, 7236515981, 10915151070941, ... 1, 1, 174, 188196, 586368681, 3745777177366, 40684710729862072, ... MAPLE A:= (n, k)-> b([n\$k]): b:= proc(l) option remember; `if`({l[]} minus {0}={}, 1, add(`if`(g(l, i), b(subsop(i=l[i]-1, l)), 0), i=1..nops(l))) end: g:= proc(l, i) local j; if l[i]<1 then return false elif l[i]>1 then for j from i+1 to nops(l) do if l[i]<=l[j] then return false elif l[j]>0 then break fi od fi; true end: seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA a[n_, k_] := b[Array[n&, k]]; b[l_] := b[l] = If[l ~Complement~ {0} == {}, 1, Sum[If[g[l, i], b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, Length[l]}]]; g[l_, i_] := Module[{j}, If[l[[i]] < 1, Return[False], If[l[[i]] > 1, For[j = i+1, j <= Length[l], j++, If[l[[i]] <= l[[j]], Return[False], If[l[[j]] > 0, Break[]]]]]]; True]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *) CROSSREFS Rows n=0-10 give: A000012, A000142, A001147, A213863, A213864, A213865, A213866, A213867, A213868, A213869, A213870. Columns k=0+1, 2-10 give: A000012, A005807(n-1) for n>0, A213873, A213874, A213875, A213876, A213877, A213878, A213871, A213872. Main diagonal gives A213862. Cf. A213276. Sequence in context: A229557 A332700 A256268 * A069777 A225816 A227655 Adjacent sequences: A213272 A213273 A213274 * A213276 A213277 A213278 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 08 2012 STATUS approved

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Last modified August 5 04:43 EDT 2024. Contains 374935 sequences. (Running on oeis4.)