A009995 Numbers with digits in strictly decreasing order. From the Macaulay expansion of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 210, 310, 320, 321, 410, 420, 421, 430, 431, 432, 510, 520, 521, 530

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full
• base
• look

Comments

There are precisely 1023 terms (corresponding to every nonempty subset of {0..9}).

A178788(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2010

A193581(a(n)) > 0 for n > 9. - Reinhard Zumkeller, Aug 10 2011

A227362(a(n)) = a(n). - Reinhard Zumkeller, Jul 09 2013

For a fixed natural number r, any natural number n has a unique "Macaulay expansion" n = C(a_r,r)+C(a_{r-1},r-1)+...+C(a_1,1) with a_r > a_{r-1} > ... > a_1 >= 0. If r=10, concatenating the digits a_r, ..., a_1 gives the present sequence. The representation is valid for all n, but the concatenation only makes sense if all the a_i are < 10. - N. J. A. Sloane, Apr 05 2014

a(n) = A262557(A263327(n)); a(A263328(n)) = A262557(n). - Reinhard Zumkeller, Oct 15 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1023
• B. Sury, Macaulay Expansion, Amer. Math. Monthly 121 (2014), no. 4, 359--360. MR3183022. [See p. 359. - _N. J. A. Sloane_, Apr 05 2014]
• Eric Weisstein's World of Mathematics, Digit.

Crossrefs

Cf. A009993.

Cf. A262557 (sorted lexicographically), A263327, A263328.

Programs

Haskell

import Data.Set (fromList, minView, insert)
a009995 n = a009995_list !! n
a009995_list = 0 : f (fromList [1..9]) where
   f s = case minView s of
         Nothing     -> []
         Just (m,s') -> m : f (foldl (flip insert) s' $
                              map (10*m +) [0..m `mod` 10 - 1])
-- Reinhard Zumkeller, Aug 10 2011

Mathematica

Sort@ Flatten@ Table[FromDigits /@ Subsets[ Range[9, 0, -1], {n}], {n, 10}] (* Zak Seidov, May 10 2006 *)

PARI

is(n)=fromdigits(vecsort(digits(n),,12))==n \\ Charles R Greathouse IV, Apr 16 2015