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

%I A009995 #54 Jul 08 2025 01:26:32
%S A009995 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,
%T A009995 61,62,63,64,65,70,71,72,73,74,75,76,80,81,82,83,84,85,86,87,90,91,92,
%U A009995 93,94,95,96,97,98,210,310,320,321,410,420,421,430,431,432,510,520,521,530
%N A009995 Numbers with digits in strictly decreasing order. From the Macaulay expansion of n.
%C A009995 There are precisely 1023 terms (corresponding to every nonempty subset of {0..9}).
%C A009995 A178788(a(n)) = 1. - _Reinhard Zumkeller_, Jun 30 2010
%C A009995 A193581(a(n)) > 0 for n > 9. - _Reinhard Zumkeller_, Aug 10 2011
%C A009995 A227362(a(n)) = a(n). - _Reinhard Zumkeller_, Jul 09 2013
%C A009995 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
%C A009995 a(n) = A262557(A263327(n)); a(A263328(n)) = A262557(n). - _Reinhard Zumkeller_, Oct 15 2015
%H A009995 Reinhard Zumkeller, <a href="/A009995/b009995.txt">Table of n, a(n) for n = 1..1023</a>
%H A009995 B. Sury, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.04.359">Macaulay Expansion</a>, Amer. Math. Monthly 121 (2014), no. 4, 359--360. MR3183022. [See p. 359. - _N. J. A. Sloane_, Apr 05 2014]
%H A009995 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Digit.html">Digit</a>.
%t A009995 Sort@ Flatten@ Table[FromDigits /@ Subsets[ Range[9, 0, -1], {n}], {n, 10}] (* _Zak Seidov_, May 10 2006 *)
%o A009995 (Haskell)
%o A009995 import Data.Set (fromList, minView, insert)
%o A009995 a009995 n = a009995_list !! n
%o A009995 a009995_list = 0 : f (fromList [1..9]) where
%o A009995    f s = case minView s of
%o A009995          Nothing     -> []
%o A009995          Just (m,s') -> m : f (foldl (flip insert) s' $
%o A009995                               map (10*m +) [0..m `mod` 10 - 1])
%o A009995 -- _Reinhard Zumkeller_, Aug 10 2011
%o A009995 (PARI) is(n)=fromdigits(vecsort(digits(n),,12))==n \\ _Charles R Greathouse IV_, Apr 16 2015
%Y A009995 Cf. A009993.
%Y A009995 Cf. A262557 (sorted lexicographically), A263327, A263328.
%K A009995 nonn,fini,full,base,look
%O A009995 1,3
%A A009995 _N. J. A. Sloane_