A product of 3 polynomials, and what its coefficients count

• at size 1 the two statistics give the same count at value 0 ×25
• the coefficient of x^0 is the number of combinations totalling 0 ×10
• the coefficient of x^0 is the number of objects of size 0 ×9
• the coefficient of xᵐ/m! at m = 0 is the number of objects on 0 labels ×8
• the coefficient of x^0 is the sum along its diagonal ×7
• row 1 adds to the plain count at that size ×6
• the row for k = 0 adds to the ordinary binomial coefficient ×6
• the row for k = 0 has degree k(n − k) ×6
• at size 1 the derivative at q = 1 is the total of the statistic over the objects ×5
• the mean number of inversions at n = 1 is n(n − 1)/4 ×5
• the bars at size 3 add to the plain count ×4
• the series is drawn to between 5 and 10 terms ×2
• a sequence of singletons on n labels is a permutation of them ×1
• an ordered pair on n labels is a two-way split of them ×1
• and the exact sign search does not find them all on the negative axis ×1
• and the marked coefficient counts the combinations written out beside it ×1
• as many sign changes on the negative axis as the degree: every root is real and negative ×1
• at least one coefficient exceeds one, so the polynomial is not merely a string of ones ×1
• between two and four boxes, each offering between two and five whole-number values ×1
• between two and four sizes, each one this statistic is enumerated at ×1
• both label-counts are short lists of small whole numbers ×1
• both series are short lists of small whole numbers ×1
• each root of unity is a root of the inversion polynomial ×1
• each root's reciprocal is a root ×1
• every combination of choices is listed ×1
• one root for each unit of degree ×1
• the checked size is between 3 and 6 ×1
• the coefficient at size 4 counts a deal of the labels together with a structure on each side ×1
• the coins give mean (n − 1)/2 and variance (n + 1)/12 ×1
• the counts rise to one peak and fall away from it ×1
• the deal is a choice rather than a formality — the ordinary product would count one ×1
• the descent row satisfies every one of Newton's inequalities ×1
• the distance to the bell falls as n grows ×1
• the first piece takes between 1 and n − 1 of them ×1
• the four rows give four different counts, so the table is separating the constructions ×1
• the inversion row fails every one ×1
• the labelled count exceeds the unlabelled one, which is what the binomial buys ×1
• the labelled family is one this family knows ×1
• the labelled object has between 3 and 6 labels ×1
• the last row is spread over more than one value, so the statistic is doing something ×1
• the marked coefficient is a small whole power ×1
• the marked coefficient is inside the product ×1
• the marked size is inside both series ×1
• the marked total is one the boxes can reach ×1
• the mean grows with the size, so the reading is not a constant ×1
• the recurrence agrees with counting descents directly ×1
• the sequence is growing, so the check has content ×1
• the sequence is one the family knows ×1
• the series inverted has a non-zero constant term ×1
• the series is drawn to between 3 and 10 terms (20 for the root views) ×1
• the sizes drawn are between 3 and 5 ×1
• the sizes drawn are ones this statistic is enumerated at ×1
• the statistic is descents or cycles ×1
• the statistic is one this family tracks ×1
• the sum of independent coins has exactly the counted distribution ×1
• the table is drawn to a size this statistic is enumerated at ×1
• the top of the coefficient is between 3 and 6 ×1
• the two statistics disagree on individual objects, so the agreement is of distributions ×1
• the value at q = 1 is the plain count ×1
• the view is one the family draws ×1
• there are C(4, 2) ways to deal the labels between the two pieces ×1
• though it is still log-concave in the plain sense ×1
• two different statistics this family tracks ×1

The whole library · What the figures prove