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 A227578 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component such that for each point (p_1,p_2,...,p_k) we have p_1<=p_2<=...<=p_k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 24
 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 14, 29, 8, 1, 1, 1, 42, 290, 185, 16, 1, 1, 1, 132, 3532, 7680, 1257, 32, 1, 1, 1, 429, 49100, 456033, 238636, 8925, 64, 1, 1, 1, 1430, 750325, 34426812, 77767945, 8285506, 65445, 128, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Conjecture: column k is asymptotic to c(k) * (k+1)^(k*n)/n^((k^2-1)/2), where c(k) is a constant dependent only on k. - Vaclav Kotesovec, Jul 21 2013 LINKS Alois P. Heinz, Antidiagonals n = 0..25, flattened Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra and its Applications, Volume 530, 1 October 2017, p. 414-444. EXAMPLE A(4,0) = 1: [()]. A(3,1) = 4: [(3),(0)], [(3),(1),(0)], [(3),(2),(0)], [(3),(2),(1),(0)]. A(2,2) = 5: [(2,2),(0,2),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)]. A(1,3) = 1: [(1,1,1),(0,1,1),(0,0,1),(0,0,0)]. A(0,4) = 1: [(0,0,0,0)]. Square array A(n,k) begins:   1,  1,    1,      1,        1,           1, ...   1,  1,    1,      1,        1,           1, ...   1,  2,    5,     14,       42,         132, ...   1,  4,   29,    290,     3532,       49100, ...   1,  8,  185,   7680,   456033,    34426812, ...   1, 16, 1257, 238636, 77767945, 36470203156, ... MAPLE b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(       i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))     end: A:= (n, k)-> `if`(k=0, 1, b([n\$k])): seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA b[l_] := b[l] = If[ l[[-1]] == 0, 1, Sum[ Sum[ b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i-1]]], l[[i]]-1}], {i, 1, Length[l]}]]; a[n_, k_] := If[k == 0, 1, b[Array[n&, k]]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *) CROSSREFS Columns k=0-10 give: A000012, A011782, A059231, A227580, A227583, A227596, A227597, A227598, A227599, A227600, A227601. Rows n=0+1, 2-10 give: A000012, A000108(k+1), A181197(k+2), A227584, A227602, A227603, A227604, A227605, A227606, A227607. Main diagonal gives: A227579. Cf. A060854 (steps decrement one component by 1), A262809, A263159. Sequence in context: A241194 A008326 A181196 * A181783 A121395 A275377 Adjacent sequences:  A227575 A227576 A227577 * A227579 A227580 A227581 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 16 2013 STATUS approved

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Last modified April 23 13:25 EDT 2021. Contains 343204 sequences. (Running on oeis4.)