## Monday, May 27, 2024

### Stable pandigital numbers

The recent OEIS A373117 addition had me wondering how many of the 3265920 pandigitals were "stable" (or "balanced" in Éric Angelini's article). I found 135914 such:

1  1023469875
2  1023487695
3  1023495876
4  1023497658
5  1023569748
6  1023578496
7  1023579468
8  1023584976
9  1023587649
10  1023596478
11  1023649857
12  1023685497
13  1023746895
14  1023748596
15  1023749568
16  1023764958
17  1023765849
18  1023845796
19  1023847659
20  1023865479
21  1024368975
...
135894  9876253014
135895  9876305421
135896  9876314250
135897  9876321540
135898  9876324051
135899  9876325104
135900  9876340251
135901  9876350214
135902  9876351024
135903  9876403512
135904  9876405132
135905  9876405213
135906  9876412503
135907  9876431052
135908  9876432105
135909  9876502413
135910  9876503142
135911  9876510432
135912  9876513024
135913  9876520314
135914  9876521043

The entire list is here. The vertical blades in the above are the seventh digit in the first 21 terms and the fourth digit in the final 21 terms. There are eighteen different ways in which ten-digit integers might be stable. Here are the possible digit-multipliers, in pandigitals the frequency of their occurrence, and the [smallest index, largest index]:

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
(-1, 1, 2, 3, 4, 5, 6, 7, 8, 9)
(-1, 0, 1, 2, 3, 4, 5, 6, 7, 8)
(-2, -1, 1, 2, 3, 4, 5, 6, 7, 8)
(-2, -1, 0, 1, 2, 3, 4, 5, 6, 7)
(-3, -2, -1, 1, 2, 3, 4, 5, 6, 7)
(-3, -2, -1, 0, 1, 2, 3, 4, 5, 6)        =>  2010  [60132, 135914]
(-4, -3, -2, -1, 1, 2, 3, 4, 5, 6)       => 14892  [11501, 135847]
(-4, -3, -2, -1, 0, 1, 2, 3, 4, 5)       => 37712  [ 4507, 135595]
(-5, -4, -3, -2, -1, 1, 2, 3, 4, 5)      => 36873  [ 1194, 134669]
(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4)      => 31768  [  345, 128927]
(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4)     => 11393  [   45, 110619]
(-6, -5, -4, -3, -2, -1, 0, 1, 2, 3)     =>  1266  [    1,  60133]
(-7, -6, -5, -4, -3, -2, -1, 1, 2, 3)
(-7, -6, -5, -4, -3, -2, -1, 0, 1, 2)
(-8, -7, -6, -5, -4, -3, -2, -1, 1, 2)
(-8, -7, -6, -5, -4, -3, -2, -1, 0, 1)
(-9, -8, -7, -6, -5, -4, -3, -2, -1, 1)