

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
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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), 976984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for nonassociative products, Bull. Amer. Math. Soc., 54 (1948), 352360.
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[2](n); nl:= min(b1, nb/2); n *aa(nl) *aa(n1nl): 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[b1, nb/2]; n*aa[nl]*aa[n1nl]]]; a[n_] := aa[n+1]/(n+1); Table[a[i], {i, 0, 50}] (* JeanFrançois 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



