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A295638 Take the sequence of nonnegative integers whose decimal digits are not in strictly increasing order. Partition the sequence into subsequences whose elements are consecutive integers. Then a(n) is the number of elements in the n-th partition.

%I A295638 #16 Feb 28 2018 03:14:21
%S A295638 2,3,4,5,6,7,8,9,33,4,5,6,7,8,9,44,5,6,7,8,9,55,6,7,8,9,66,7,8,9,77,8,
%T A295638 9,88,9,99,444,5,6,7,8,9,55,6,7,8,9,66,7,8,9,77,8,9,88,9,99,555,6,7,8,
%U A295638 9,66,7,8,9,77,8,9,88,9,99,666,7,8
%N A295638 Take the sequence of nonnegative integers whose decimal digits are not in strictly increasing order. Partition the sequence into subsequences whose elements are consecutive integers. Then a(n) is the number of elements in the n-th partition.
%C A295638 Only defined as an integer for a(1) through a(255), as a(256) references the infinite partition (123456790, 123456791, ..., 999999999, 1000000000, 1000000001, ...). No integer greater than 123456789 has a strictly increasing sequence of digits (itself being the only case for 9 digits, and by the pigeonhole principle, a >9-digit number must have a digit repeated and is thus not strictly increasing).
%H A295638 Gunnar Lee Johnson, <a href="/A295638/b295638.txt">Table of n, a(n) for n = 1..255</a>
%e A295638 For a(1)=2 through a(8)=9, these correspond to the consecutive subsequences (10, 11), (20, 21, 22), ..., (80, 81, 82, ..., 88). The jumps at e.g. a(9)=33 or a(37)=444 correspond to (90, 91, ..., 122) and (790, 791, ..., 1233), where 89 and 123, and 789 and 1234, are the values partitioning the subsequences.
%o A295638 (Python) def a(n):
%o A295638     (x,i,count,switch) = (0,0,1,True)
%o A295638     while True:
%o A295638         if switch == (list(sorted(set(str(i)))) == list(str(i))):
%o A295638             count += 1
%o A295638         else:
%o A295638             if not switch: x += 1
%o A295638             if x == n: return count
%o A295638             (count, switch) = (1, not switch)
%o A295638         i += 1
%o A295638 (PARI) is(n) = my(d=digits(n)); d != vecsort(d,,8);
%o A295638 lista(nn) = {my(w = select(n->is(n), vector(nn, k, k))); my(dw = vector(#w-1, k, w[k+1] - w[k])); my(k = 1); for (n=1, #dw, if (dw[n] == 1, k++, print1(k, ", "); k = 1););} \\ _Michel Marcus_, Jan 08 2018
%Y A295638 The nonnegative integers minus A009993 is the sequence that is partitioned.
%K A295638 nonn,fini,less,base
%O A295638 1,1
%A A295638 _Gunnar Lee Johnson_, Nov 24 2017