A182220 Largest number k such that there exists an extensional acyclic digraph on n labeled nodes with k sources.

1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 58, 59, 60, 61

Offset: 1

Author

Nathaniel Johnston, Apr 19 2012

Keywords

• nonn

Comments

Also the length of row n of A182162.

This seems to be simply the natural numbers, with the terms in A000325 repeated.

It appears a(n+1) is the number of distinct possible heights of binary trees with n nodes. The minimum height of an n node binary tree is A000523(n), the maximum height is n-1 and all intermediate heights are possible. This conjecture is therefore equivalent to the conjectured formulas. - Yuchun Ji, Mar 22 2021

Conjecture: Partial sums of A347523, thus a(n) is the number of nonpowers of 2 <= n-1, or with offset 0: a(n) is the number of nonpowers of 2 <= n. - Omar E. Pol, Sep 30 2021

Links

• S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).

Crossrefs

Cf. A083058, A182162.

Programs

Maple

A001192 := proc(n) option remember: if(n=0)then return 1: fi: return add((-1)^(n-k-1)*binomial(2^k-k, n-k)*procname(k), k=0..n-1); end: A182162 := proc(n, l) local vl: vl := add((-1)^(k-l)*binomial(n, k)*binomial(k, l)*binomial(2^(n-k)-n+k, k)*k!*(n-k)!*A001192(n-k), k=l..n): return vl: end: A182220 := proc(n) local l: for l from n to 1 by -1 do if(A182162(n, l)>0)then break:fi:od: return l: end: seq(A182220(n),n=1..60);

Formula

Conjecture, for all n >= 3: a(n) = A083058(n-1) + 1 = n - 1 - A000523(n-1) = n - 1 - floor(log(2,n)). - Antti Karttunen, Aug 17 2013

Conjecture: a(1) = 0, a(n) = n - 1 - Sum_{i=1..n} sign(floor((n-1)/ 2^i)), n > 1. - Wesley Ivan Hurt, Feb 02 2014

Conjecture: a(n) = n - Sum_{k=0..n-2} A036987(k). - Paul Barry, Mar 07 2017