A214283 Smallest Euler characteristic of a downset on an n-dimensional cube.

0, -1, -2, -3, -4, -10, -20, -35, -56, -126, -252, -462, -792, -1716, -3432, -6435, -11440, -24310, -48620, -92378, -167960, -352716, -705432, -1352078, -2496144, -5200300, -10400600, -20058300, -37442160, -77558760, -155117520, -300540195

Offset: 1

Author

Terence Tao, Jul 09 2012

Keywords

• sign

Comments

An m-downset is a set of subsets of 1..m such that if S is in the set, so are all subsets of S. The Euler characteristic of a downset is the number of sets in the downset with an even cardinality, minus the number with an odd cardinality.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Terry Tao, Optimal bounds for an alternating sum on a downset, 2012.

Crossrefs

Cf. A000108, A001405, A004525, A007318, A212831, A214282.

Programs

Haskell

a214283 1 = 0
a214283 n = - a007318 (n - 1) (a004525 n)
-- Reinhard Zumkeller, Jul 14 2012

Mathematica

A212831[n_] := (1/4)*((-(1+(-1)^n))*(-1+(-1)^Floor[n/2]) - (-3+(-1)^n)*n); a[n_] := -CatalanNumber[Floor[(n-1)/2]]*A212831[n-1]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Nov 06 2012, after Paul Curtz *)

PARI

a(n)=-binomial(n-1,if(n%2,if(n%4==3,n-1,n+1),n)/2) \\ Charles R Greathouse IV, Jul 10 2012

Python

from math import comb
def A214283(n): return -comb(n-1,(n>>1)|(n&1)) # Chai Wah Wu, Jan 31 2024

Formula

a(n=2k) = -binomial(n-1,n/2) = -binomial(2k-1,k),

a(n=4k+3) = -binomial(n-1,(n-1)/2) = -binomial(4k+2,2k+1),

a(n=4k+1) = -binomial(n-1,(n+1)/2) = -binomial(4k,2k+1).

a(n) = A214282(n) - A001405(n). - Reinhard Zumkeller, Jul 14 2012

For n > 1: a(n) = - A007318(n-1, A004525(n)). - Reinhard Zumkeller, Jul 14 2012

a(n+1) = -A000108(n/2) * A212831(n). - Paul Curtz, Nov 04 2012