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 A133385 Number of permutations of n elements divided by the number of (binary) heaps on n+1 elements. 3
 1, 1, 1, 2, 3, 6, 9, 24, 45, 108, 189, 504, 945, 2268, 3969, 12096, 25515, 68040, 130977, 381024, 773955, 2000376, 3750705, 11430720, 24111675, 64297800, 123773265, 360067680, 731387475, 1890355320, 3544416225, 11522165760, 25823603925 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS In a heap on (n+1) distinct elements only n elements can change places, since the first element is determined to be the minimum. a(n) gives the number of all possibilities divided by the number of legal possibilities to do this. Is this the sequence mentioned on page 360 of Motzkin (1948)? - N. J. A. Sloane, Jul 04 2015 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984. T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360. Eric Weisstein's World of Mathematics, Heap Wikipedia, Binary heap FORMULA a(n) = A132862(n+1)/(n+1) = A000142(n)/A056971(n+1). EXAMPLE a(4)=3 because 3=24/8 and there are 4!=24 permutations on 4 elements and 8 heaps on 5 elements, namely (1,2,3,4,5), (1,2,3,5,4), (1,2,4,3,5), (1,2,4,5,3), (1,2,5,3,4), (1,2,5,4,3), (1,3,2,4,5) and (1,3,2,5,4). In every (min-) heap, the element at position i has to be larger than an element at position floor(i/2) for all i=2..n. The minimum is always found at position 1. MAPLE aa:= proc(n) option remember; local b, nl; if n<2 then 1 else b:= 2^ilog(n); nl:= min(b-1, n-b/2); n *aa(nl) *aa(n-1-nl): fi end: a:= n-> aa(n+1)/(n+1): seq(a(i), i=0..50); MATHEMATICA aa[n_] := aa[n] = Module[{b, nl}, If[n<2, 1, b = 2^Floor[Log[2, n]]; nl = Min[b-1, n-b/2]; n*aa[nl]*aa[n-1-nl]]]; a[n_] := aa[n+1]/(n+1); Table[a[i], {i, 0, 50}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *) CROSSREFS Cf. A000142, A056971, A132862. Column k=2 of A273730. Sequence in context: A323144 A056353 A111274 * A002076 A286435 A145761 Adjacent sequences:  A133382 A133383 A133384 * A133386 A133387 A133388 KEYWORD nonn AUTHOR Alois P. Heinz, Nov 22 2007 STATUS approved

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Last modified July 11 01:51 EDT 2020. Contains 335600 sequences. (Running on oeis4.)