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 A261835 Number A(n,k) of compositions of n into distinct parts where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 16, 3, 0, 1, 5, 10, 46, 21, 5, 0, 1, 6, 15, 100, 75, 50, 11, 0, 1, 7, 21, 185, 195, 231, 205, 13, 0, 1, 8, 28, 308, 420, 736, 1414, 292, 19, 0, 1, 9, 36, 476, 798, 1876, 6032, 2376, 587, 27, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Also matrices with k rows of nonnegative integers with distinct positive column sums and total element sum n. A(2,2) = 3: (matrices and corresponding marked compositions are given)   [1]   [2]   [0]   [1]   [0]   [2]   2ab,  2aa,  2bb. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,k) = Sum_{i=0..k} C(k,i) * A261836(n,k-i). EXAMPLE A(3,2) = 16: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb. Square array A(n,k) begins:   1,  1,   1,    1,     1,     1,      1,      1, ...   0,  1,   2,    3,     4,     5,      6,      7, ...   0,  1,   3,    6,    10,    15,     21,     28, ...   0,  3,  16,   46,   100,   185,    308,    476, ...   0,  3,  21,   75,   195,   420,    798,   1386, ...   0,  5,  50,  231,   736,  1876,   4116,   8106, ...   0, 11, 205, 1414,  6032, 19320,  51114, 117936, ...   0, 13, 292, 2376, 11712, 42610, 126288, 322764, ... MAPLE b:= proc(n, i, p, k) option remember;       `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))     end: A:= (n, k)-> b(n\$2, 0, k): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2 < n, 0, If[n == 0, p!, b[n, i-1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; A[n_, k_] := b[n, n, 0, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 16 2017, translated from Maple *) CROSSREFS Columns k=0-10 give: A000007, A032020, A261840, A261841, A261842, A261843, A261844, A261845, A261846, A261847, A261848. Rows n=0-4 give: A000012, A001477, A000217, A255211, A228317(n+2). Main diagonal gives A261837. Cf. A261780, A261836. Sequence in context: A286335 A291652 A071569 * A286932 A259475 A323224 Adjacent sequences:  A261832 A261833 A261834 * A261836 A261837 A261838 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 02 2015 STATUS approved

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Last modified September 18 12:28 EDT 2020. Contains 337169 sequences. (Running on oeis4.)