

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 nth partition.


1



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, 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, 9, 66, 7, 8, 9, 77, 8, 9, 88, 9, 99, 666, 7, 8
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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 >9digit number must have a digit repeated and is thus not strictly increasing).


LINKS

Gunnar Lee Johnson, Table of n, a(n) for n = 1..255


EXAMPLE

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.


PROG

(Python) def a(n):
(x, i, count, switch) = (0, 0, 1, True)
while True:
if switch == (list(sorted(set(str(i)))) == list(str(i))):
count += 1
else:
if not switch: x += 1
if x == n: return count
(count, switch) = (1, not switch)
i += 1
(PARI) is(n) = my(d=digits(n)); d != vecsort(d, , 8);
lista(nn) = {my(w = select(n>is(n), vector(nn, k, k))); my(dw = vector(#w1, 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


CROSSREFS

The nonnegative integers minus A009993 is the sequence that is partitioned.
Sequence in context: A349864 A004849 A261279 * A357195 A173551 A271534
Adjacent sequences: A295635 A295636 A295637 * A295639 A295640 A295641


KEYWORD

nonn,fini,less,base


AUTHOR

Gunnar Lee Johnson, Nov 24 2017


STATUS

approved



